GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 1, 1997, 41-64

HARMONIC MAPS OVER RINGS

A. LASHKHI

Abstract. For the torsion-free modules over noncommutative principal ideal domains von Staudt’s theorem is proved. Moreover, more general (nonbijective) harmonic maps with the classical definition of harmonic quadruple is calculated.

Introduction

K. von Staudt stated a theorem which clearly shows that it is important to consider the manner in which the blocks are embedded in order to get information on the surrounding geometrical structure. It could be considered as the spring of the geometric algebra.

The modern flavor of the subject was established by E. Artin [1], R. Baer [2], and J. Dieudonné [3]. These classic studies described the theory over division rings. R. Baer and J. von Neumann pointed out a possible extension of the structural identity between (projective) geometry and linear algebra to the case of a ring, generating intense research activity in the area of geometric algebra over rings. The main problem in this field is to translate the specific maps from the geometrical point of view (perspectivities. collineations, harmonic maps) in algebraic language (by the semilinear isomohphisms) – the fundamental theorems of geometric algebra. A continuing investigation by many scholars over the last 30 years has charted the evolution of the classical setting into a stable form for the general rings. NATO ASI held conferences twice on the subject and published two books [4], [5].

The boundaries of the subject “What is geometric algebra?” were established by Artin, Baer, and Dieudonné. These classic studies described the structure theory, actions, transitivity, normal subgroups, commutators and automorphisms of the classical linear groups (general linear, symplectic, orthogonal, unitary) from the geometrical point of view.

41

1991 Mathematics Subject Classification. 51A10.

Key words and phrases. Projective space, harmonic map, collineation.

1072-947X/97/0100-0041$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

On the other hand this field of problems is central in the isomorphism theory of the classical groups, which gave way to an extensive isomorphism theory of certain full classical groups. The isomorphism GL_{n}(K) → GL_{n}(K_{1}) in turn gives rise to isomorphisms between the corresponding elementary groups and the (projective) geometric versions of these groups.

What is the fundamental theorem of geometric algebra? For different geometries it can be stated in various ways. However, in general, the problem is to represent specific geometrical maps by the linear functions, i.e., with the elements of GL(k, X), where X is a k-module. In the classical case when k = F is a field or division ring the following approximate versions of the representations are well known:

(P_{1}) Perspectivities by linear maps + trivial automorphism of F; (P_{2}) Collineations by linear maps + automorphism of F; (P_{3}) Harmonic maps by linear maps + automorphism or antiautomor-

phism of F.

Naturally, for different geometries (affine, projective, symplectic, orthogonal, unitary, etc.) all the above-mentioned versions have a specific flavor. The most developed ring version is the projective case. Recall that the projective geometry P G(k, X) of a torsion-free k-module X can be realized as the lattice of all k-free submodules. In this direction the most significant result is Ojanguren and Sridharans’ theorem which generalizes to commutative rings the classical theorem of projective geometry [6].

These and some later results give us a reason to suppose that the theorem of type (P_{3}) is true for some general noncommutative rings.

Let us formulate the theorem (K. von Staudt’s theorem) for the classical case, i.e., for vector spaces over skew fields.

Theorem A (Case dim_{p} A = 1). Let X and X_{1} be vector spaces over the skew fields F and F_{1}, respectively, dim_{p} X = dim_{p} X_{1} = 1. Let f : P (X) ⟶ P (X_{1}) be some map. Then the following alternatives are equivalent:

(a) f is bijective and harmonic;

(b) f is bijective and f, f^{−1} are harmonic;

(c) f is a nontrivial harmonic map;

(d) f is bijective and harmonic for the fixed quadruple;

(e) there exists either an isomorphism or an anti-isomorphism σ : F ⟶ F_{1} and a σ-semilinear isomorphism μ : X ⟶ X_{1} such that f(F x) = F_{1}μ(x) for all x ∈ X.

The definition of a semilinear isomorphism with respect to the antiisomorphism will be given later (Definition 5).

Naturally, for general rings the conditions (a)–(d) are not equivalent and we get the following implications:

[figure: (d); (b); (a); (c)]

To prove the fundamental theorem means to show the validity of (e) from one of the conditions (a)–(d).

When trying to extend the concepts of (projective) geometry for a given ring, the following question arises: what is the projective space? It can be defined in two ways:

(i) P̃(X) as the set of all k-free direct summands of rank 1.

(ii) P (X) as the set of all k-free submodules of rank 1.

It is known that P̃(X) does not always give the desired results. However, taking into consideration P (X), we can get positive results for some general noncommutative rings.

The first generalization of von Staudt’s theorem belongs to G. Ancochea [7]. In spite of the foregoing theorem one can extend K. von Staudt’s theorem to some special commutative rings, in particular, if k is a commutative local or semilocal ring (N. B. Limaye [8], [9]), or if k is a commutative algebra of finite dimension over a field of sufficiently large order (H. Schaeffer [10]), or if k is a commutative primitive ring (B. R. McDonald [11]). Furthermore, B. V. Limaye and N. B. Limaye [12] generalized the theorem to noncommutative local rings by adopting the definition of a harmonic map. However, for commutative principal ideal domains the Staudt’s theorem is invalid [13], [14].

W. Klingenberg in 1956 introduced the idea of “non-injective collineations” between projective spaces of two and three dimensions. In a series of papers F. Veldkamp (partly together with J. C. Ferrar) developed the theory of homomorphisms of ring geometry, which are, roughly speaking, noninjective collineations [15], [16], [17], [18]. The first article of non-injective harmonic maps between projective lines was due to F. Buekenhout [19], after D. G. James [20] got the same result. Buekenhout’s work described the situation for division rings. In 1985 C. Bartolone and F. Bartolozzi extended some of Buekenhout’s ideas for the ring case [14].

Cross-ratio, harmonic quadruple, and von Staudt’s theorem in Moufang planes was studied by V. Havel [21], [22] and J. C. Ferrar [23].

Many interesting and fundamental results according, this and boundary problems were obtained by W. Benz and his scholars [10], [24]–[26]. A. Dress and W. Wenzel constitute an important tool of cross-ratios from a combinatorial point of view [27].

Some other generalizations and related problems can be found in [28]– [41]. For more complete information and an exhaustive bibliography in this area see [14], [28], [35].

Our aim is to calculate more general (nonbijective) harmonic maps satisfying the condition (c) with the classical definition of a harmonic quadruple for some general noncommutative rings and to obtain a complete analog of the classical case. Moreover, using some ideas from [37] we’ll consider not only free modules but also the torsion-free ones and calculate σ and μ (i.e., semilinear isomorphism) having given f, X, k, P (X), P G(k, X).

The notions and definitions are standard. k is an arbitrary integral domain with unity; all modules are over k; P G(k, X) is the projective geometry of the k-module X, i.e., the lattice of all k-free submodules; 𝔐(X) is the complete lattice of all submodules X; ⟨Y ⟩ denotes the submodule generated by the set Y. Note also that to fix the basic ring k sometimes we’ll write P_{k}(X) and 𝔐_{k}(X).

1. Projective Space, Collineation and Cross-Ratio

Let k be a commutative ring with unit. For each k-free module X we can construct a new object (see [6], [14]–[18], [35]), the projective space P̃(X) corresponding to X. The elements of P̃(X) are k-free direct summands of rank 1. It is clear that each element of P̃(X) has the form ke, i.e., is a one-dimensional submodule generated by the unimodular element e ∈ X. Remember that an element e is unimodular if there exists a linear form μ : X ⟶ k such that μ(e) = 1, i.e., the coordinates of e in one of the bases X generate the unit ideal of k. If e_{1}, e_{2}, …, e_{n}, … is a basis of the k-module X, then e = ∑ a_{i}e_{i} is unimodular if and only if ∑ ka_{i} = k. This definition of the projective space we widen in the following way:

Definition 1. Let k be an integral domain (not necessarily commutative). X is a torsion-free module over k. The projective space P (X) corresponding to X is the set of all k-free submodules of rank 1.

Note that Definition 1 is meaningful for every torsion-free module X and it can happen that for some k-module X, P̃(X) = ∅ while P (X) ≠ ∅. It is also obvious that if U ⊂ X is a submodule, then e is unimodular in U while e is not unimodular in X. For every k-free submodule U ↪ X the projective dimension dim_{p} will be defined as dim_{p} U = dim U − 1. We shall use the terms: “point”, “line”, “plane” for free submodules of the projective dimensions 0,1,2. We shall condider the zero submodule as an “empty element” of the projective space P (X) with projective dimension −1.

Definition 2. The set of points {P_{α}, α ∈ Λ} of the projective space P (X) will be called collinear, if there exists a line U ↪ X such that P_{α} ∈ U for every α ∈ Λ and strictly collinear if there exists a line U for which U = P_{α} + P_{β}, for every α, β ∈ Λ.

If the set of points is strictly collinear, then the line U will be called the principal line passing through these points.

In the sequel k^{∗} is the group of units of the ring k. If s ∈ k is an arbitrary element, then by [s] we denote the set of conjugate elements of the form t^{−1}st, where t ∈ k^{∗}.

The points P, Q ∈ P (X) are independent if P ∩ Q = 0 and dependent if P ∩ Q ≠ 0.

Let P_{1} = ke_{1}, P_{2} = ke_{2} be independent. If U = P_{1} + P_{2} and P_{3} = k(αe_{1} + βe_{2}) ↪ U is an arbitrary point, then it is obvious that the points P_{1}, P_{2}, P_{3} are strictly collinear if and only if α, β ∈ k^{∗}. It is also obvious that if P_{1}, P_{2}, P_{3}, P_{4} are strictly collinear points and U is a principal line passing through these points, then there exist unimodular elements e_{1}, e_{2} of this line U and an invertible element s ∈ k^{∗} such that

P_{1} = ke_{1}, P_{2} = ke_{2}, P_{3} = k(e_{1} + e_{2}), P_{4} = k(e_{1} + se_{2}).

The element s ∈ k^{∗} is called the cross-ratio of these points. If k is commutative, then s is unique. For the non-commutative situation the cross-ratio is [s]. For s′ = tst^{−1} we have

P_{1} = k(te_{1}), P_{2} = k(te_{2}), P_{3} = k(te_{1} + te_{2}), P_{4} = k(te_{1} + s′(te_{2})).

On the other hand, if

P_{1} = k\overline{e}_{1}, P_{2} = k\overline{e}_{2}, P_{3} = k(\overline{e}_{1} + \overline{e}_{2}), P_{4} = k(\overline{e}_{1} + s_{1}\overline{e}_{2}),

then we have

P_{1} = k\overline{e}_{1}, P_{2} = k\overline{e}_{2}, P_{3} = k(\overline{e}_{1} + \overline{e}_{2}), P_{=}k(\overline{e}_{1} + s′te_{2})),
\overline{e}_{1} = μ_{1}e_{1}, \overline{e}_{2} = μ_{2}e_{2}, \overline{e}_{1} + \overline{e}_{2} = μ_{3}(e_{1} + e_{2}),
\overline{e}_{1} + s′\overline{e}_{2} = μ_{4}(e_{1} + se_{2}) ⟹ μ_{1}e_{1} + μ_{2}e_{2} = μ_{3}(e_{1} + e_{2}),
μ_{1} = μ_{2} = μ_{3} ⟹ μ_{1}e_{1} + s′μ_{2}e_{2} = μ_{4}e_{1} + μ_{4}se_{2},
μ_{1} = μ_{4} ⟹ s′μ_{4} = μ_{4}s.

For the quadruple of strictly collinear points and their cross-ratio we use the notation

[P_{1}, P_{2}, P_{3}, P_{4}] = [s].

We remark that the order of the points P_{i} is essential.

Let now e_{1} and e_{2} be generators of a k-free submodule U of rank 2. Consider the points k(α_{i}e_{1} + β_{i}e_{2}), α_{i}, β_{i} ∈ k^{∗}, 1 ≤ i ≤ 4. For i ≠ j we

shall use the notation

D_{ij} =|\begin{matrix}α_{i}, & β_{i} \\ α_{j}, & β_{j} \end{matrix}| = α_{i}β_{j} − α_{j}β_{i}; D̃_{ij} =\begin{matrix} \\ | & \begin{matrix}α_{i}, & β_{i} \\ α_{j}, & β_{j} \end{matrix} & | \end{matrix} = α_{i}α_{j}^{−1} − β_{i}β_{j}^{−1}.

Proposition 1. The points k(α_{i}e_{i} + β_{i}e_{2}) are strictly collinear if and only if

D_{ij} ∈ k^{∗},  D̃_{ij} ∈ k^{∗}.

Proof. We have

β_{i}^{−1}(α_{i}e_{1} + β_{i}e_{2}) = β_{i}^{−1}α_{i}e_{1} + e_{2} = \overline{e}_{1},
β_{j}^{−1}(α_{j}e_{1} + β_{j}e_{2}) = β_{j}^{−1}α_{j}e_{1} + e_{2} = \overline{e}_{2};
\overline{e}_{1} − \overline{e}_{2} = (β_{i}^{−1}α_{i} − β_{j}^{−1}α_{j})e_{1},
β_{i}(β_{i}^{−1}α_{i} − β_{j}^{−1}α_{j})α_{j}^{−1} = α_{i}α_{j}^{−1} − β_{i}β_{j}^{−1} ∈ k^{∗}
⟹ e_{1}, e_{2} ∈ k(α_{i}e_{1} + β_{i}e_{2}) + k(e_{j}e_{1} + β_{j}e_{2}).

The inclusion D_{ij} ∈ k^{∗} is proved straightforward.

Proposition 2. If P_{1}, P_{2}, P_{3}, P_{4} are strictly collinear points and α_{1} = 0, β_{1} = α_{2} = α_{3} = α_{4} = 1, then

D_{32}D_{42}^{−1} ∈ [P_{1}, P_{2}, P_{3}, P_{4}].

Proof.

k(e_{1} + β_{3}e_{2}) = k[(β_{3} − β_{2})e_{2} + (e_{1} + β_{2}e_{2})],
k(e_{1} + β_{4}e_{2}) = k(β_{4}e_{2} − β_{2}e_{2} + e_{1} + β_{2}e_{2}
= k[(β_{4} − β_{2}) + e_{2} + e_{1} + β_{2}e_{2}]
= k[e_{2} + (β_{4} − β_{2})^{−1}(e_{1} + β_{2}e_{2})]
= k[(β_{3} − β_{2})e_{2} + (β_{3} − β_{2})(β_{4} − β_{2})^{−1}(e_{1} + β_{2}e_{2})].

Proposition 3. If in Proposition 2, α_{1} = α_{2} = α_{3} = α_{4} = 1, then

D_{41}D_{42}^{−1}D_{32}D_{31}^{−1} ∈ [P_{1}, P_{2}, P_{3}, P_{4}].

Proof. Suppose that

e_{1} + β_{3}e_{2} = λ_{1}(e_{1} + β_{1}e_{2}) + λ_{2}(e_{1} + β_{2}e_{2})

⟹ \{ \begin{matrix}λ_{1} + λ_{2} = 1 \\ λ_{1}β_{1} + λ_{2}β_{2} = β_{3} \end{matrix} ⟹ \{ \begin{matrix}λ_{1} = 1 − λ_{2} \\ λ_{1}β_{1} + λ_{2}β_{2} = β_{3} \end{matrix}

⟹ β_{1} + λ_{2}(β_{2} − β_{1}) = β_{3}.

Consequently, λ_{2} = D_{13}D_{12}^{−1}. In the same way λ_{1} = D_{23}D_{12}^{−1}. From Proposition 1 we have β_{i} − β_{j} ∈ k^{∗} and 1 − β_{j} ∈ k^{∗}. In our conditions for 1 ≤ i, j, k ≤ 4 we have

(β_{i} − β_{j})(β_{k} − β_{j})^{−1} − 1 = (β_{i} − β_{j} − β_{k} + β_{j})(β_{k} − β_{j})^{−1}
= (β_{i} − β_{k})(β_{k} − β_{j})^{−1};

(β_{i} − β_{j})(β_{k} − β_{j})^{−1}β_{k} − β_{i} = (β_{i} − β_{j})(β_{k} − β_{j})^{−1}β_{k} − β_{k} − (β_{i} − β_{k})

= [(β_{i} − β_{j})(β_{k} − β_{j})^{−1} − 1]β_{k} − (β_{i} − β_{k})

= (β_{i} − β_{k})(β_{k} − β_{j})^{−1}β_{k} − (β_{i} − β_{k})
= (β_{i} − β_{j})(β_{k} − β_{j})^{−1}[β_{k} − (β_{k} − β_{j})]
= (β_{i} − β_{j})(β_{k} − β_{j})^{−1}β_{j}.

Taking into account these equations, we find

k[−(e_{1} + β_{1}e_{2}) + (β_{1} − β_{3})(β_{2} − β_{3})^{−1}(e_{1} + β_{2}e_{2})]

= k[((β_{1} − β_{3})(β_{2} − β_{3})^{−1} − 1)e_{1} + [(β_{1} − β_{3})(β_{2} − β_{3})^{−1}β_{2} − β_{1}]e_{2}]

= k[(β_{1} − β_{2})(β_{2} − β_{3})^{−1}e_{1} + (β_{1} − β_{2})(β_{2} − β_{3})^{−1}β_{3}e_{2}]

= k(β_{1} − β_{2})(β_{2} − β_{3})^{−1}(e_{1} + β_{3}e_{2})]

= k[e_{1} + β_{3}e_{2}] = P_{3};

k[−(e_{1} + β_{1}e_{2}) + (β_{1} − β_{4})(β_{2} − β_{4})^{−1}(e_{1} + β_{2}e_{2})]

= k[(−1+(β_{1}−β_{4})(β_{2}−β_{4})^{−1})e_{1}+[(β_{1}−β_{4})(β_{2}−β_{4})^{−1}β_{2}−β_{1}]e_{2}]

= k[(β_{1} − β_{2})(β_{4} − β_{2})^{−1}e_{1} + (β_{1} − β_{2})(β_{4} − β_{2})^{−1}β_{4}e_{2}]

= k(e_{1} + β_{4}e_{2}) = P_{4} = k[−(e_{1} + β_{1}e_{2})

+ (β_{1} − β_{4})(β_{2} − β_{4})^{−1}(β_{2} − β_{3})(β_{1} − β_{3})^{−1}(β_{2} − β_{3})^{−1}(e_{1} + β_{2}e_{2})]

= k[−(e_{1} + β_{1}e_{2}) + D_{41}D_{42}^{−1}D_{32}D_{31}^{−1}(β_{1} − β_{3})(β_{2} − β_{3})^{−1}(e_{1} + β_{2}e_{2})].

Consequently, the equations

P_{1} = k[−(e_{1} + β_{1}e_{2})],
P_{2} = k[(β_{1} − β_{3})(β_{2} − β_{3})^{−1}(e_{1} + β_{2}e_{2})]

complete the proof.

The set k^{∗} ⊂ k splits in equivalent classes of conjugate elements. Then for each class [s_{α}] on the line U = P_{1} ∪ P_{2} = P_{1} ∪ P_{3} = P_{2} ∪ P_{3} we can always find the point P_{4} such that [P_{1}, P_{2}, P_{3}, P_{4}] = [s]. In fact, we can find basic elements e_{1}, e_{2} ∈ U such that P_{1} = ke_{1}, P_{2} = ke_{2}, P_{3} = k(e_{1} + e_{2}) and then choose the point P_{4}. The point P_{4} is not uniquely defined by the points P_{1}, P_{2}, P_{3} and the cross-ratio. If the element s belongs to the center of the ring k, then P_{4} is unique, which is easy to check by straightforward calculations.

Let P_{1}, P_{2}, P_{3}, P_{4} be a quadruple of strictly collinear points on the projective line U. Then the cross-ratio depends on the order of the points. The effect of inversion is illustrated by the equations (see [2])

[P_{1}, P_{2}, P_{3}, P_{4}] = [P_{2}, P_{1}, P_{3}, P_{4}]^{−1} = [P_{1}, P_{2}, P_{4}, P_{3}]^{−1},
[P_{1}, P_{2}, P_{3}, P_{4}] = 1 − [P_{1}, P_{3}, P_{2}, P_{4}].

Note that if A ⊂ k is a subset then A^{−1} =^{def} {x^{−1}, for all x ∈ A}. The first two equations we can check from Proposition 3. The generator of the point can always be chosen in such a way that the coefficient of e_{1} will be 1. Let

P_{1} = ke_{1}, P_{2} = ke_{2}, P_{3} = k(e_{1} + e_{2}), [P_{1}, P_{2}, P_{3}, P_{4}] = [s].

Choose the basis {e_{1}, e_{2}} on {−e_{1}, e_{1} + e_{2}}. Then

e_{2} = −e_{1} + (e_{1} + e_{2}), se_{1} + e_{2} = (1 − s)(−e_{1}) + (e_{1} + e_{2})
⟹ [P_{1}, P_{2}, P_{3}, P_{4}] = 1 − [s].

The quadruple of the strictly collinear points P_{1}, P_{2}, P_{3}, P_{4} ∈ P (X) is in a harmonic relation if [P_{1}, P_{2}, P_{3}, P_{4}] = −1. Note that this definition implies that \frac{1}{2} ∈ k.

Proposition 4. Let X_{1} and X_{2} be torsion-free modules over the rings k_{1} and k_{2}; α : P (X_{1}) ⟶ P (X_{2}) = 2 be a bijection, and rank X_{1} = rank X_{2}; then the following statements are equivalent:

(a) P_{1}, P_{2}, P_{3}, P_{4} ∈ P (X_{1}) are harmonic if and only if α(P_{1}), α(P_{2}), α(P_{3}), α(P_{4}) are harmonic;

(b) if P_{1}, P_{2}, P_{3}, P_{4} are harmonic, then α(P_{1}), α(P_{2}), α(P_{3}), α(P_{4}) are harmonic, and if Q_{1}, Q_{2}, Q_{3}, Q_{4} ∈ P (X_{2}) are harmonic, then α^{−1}(Q_{1}), α^{−1}(Q_{2}), α^{−1}(Q_{3}), α^{−1}(Q_{4}) are strictly collinear.

Proof. (a)⟹ (b) is obvious. (b) ⟹ (a). Let {e_{1}, e_{2}} and {f_{1}, f_{2}} be bases of the lines U = Q_{i} + Q_{j} ⊆ X_{2} and α^{−1}(U) = α^{−1}(Q_{i}) + α^{−1}(Q_{j}), 1 ≤ i, j ≤ 4. Suppose that

α(k_{1}e_{1}) = Q_{1} = k_{2}f_{1}, α(k_{1}e_{2}) = Q_{2} = k_{2}f_{2},
α(k_{1}(e_{1} + e_{2})) = Q_{3} = k_{2}(f_{1} + f_{2}),
α(k_{1}(e_{1} + μe_{2})) = Q_{4} = k_{2}(f_{1} − f_{2}).

It is clear that μ ∈ k^{∗}. Since the triple of the strictly collinear points Q_{1}, Q_{2}, Q_{3} represents the fourth harmonic point Q_{4}, we have

α(k_{1}(e_{1} − e_{2})) = Q_{4} = α(k_{1}(e_{1} + μe_{2})) ⟹ μ = −1.

The map f : P (X_{1}) ⟶ P (X_{2}) will be called harmonic if the images of harmonic points are harmonic. f : P (X_{1}) ⟶ 𝔐(X_{2}) will be called a collineation if P_{1} ⊂ P_{2} + P_{3} implies f(P_{1}) ⊂ f(P_{2}) + f(P_{3}). The map f preserves linear independence if P_{1}, …, P_{α} ∈ P (X_{1}) are independent if and only if f(P_{1}), …, f(P_{α}) ∈ 𝔐(X_{2}) are independent, i.e., for every β ∈ Λ we have

P_{β} ∩ ( ⋃_{γ∈Λ,γ≠β} P_{γ} ) = 0 ⟺ f(P_{β}) ∩ ( ⋃_{γ∈Λ,γ≠β} f(P_{γ})) = 0.

A collineation which preserves linear independence will be called an LIPcollineation.

Let e be the unimodular element in the k-free submodule A. Then k_{1}e ⊂ ∑_{i=1}^{m} k_{1}e_{i}, where {e_{i}, i = 1, …, m} is some finite subset of the basis A. It is obvious that if f is a collineation, then f(k_{1}e) ⊂ ∑_{i=1}^{m} f(k_{1}e_{i}).

Recall that the 1–1 map f : X_{1} ⟶ X_{2} is a semilinear (σ-semilinear) isomorphism with respect to σ if σ : k_{1} ⟶ k_{2} is a ring isomorphism and

f(ax_{1} + bx_{2}) = σ(a)f(x_{1}) + σ(b)f(x_{2})

for each a, b ∈ k_{1}, x_{1}, x_{2} ∈ X_{2}.

Let U ⊆ X_{1} be a k_{1}-free submodule; f : X_{1} ⟶ X_{2} be a σ-semilinear map. It is clear that the image of the unimodular element e ∈ U is unimodular. So we get an induced map, i.e., the projection P (f) : P (X_{1}) ⟶ P (X_{2}), for which P (f)(k_{1}e) = k_{2}f(e) for all unimodular elements of all lines of X_{1}. It is also obvious that P_{1} ⊂ P_{2} +P_{3} implies P (f)P_{1} ⊂ P (f)P_{2} +P (f)P_{3}.

2. Some Facts Concerning Harmonic Maps and Collineations

Let k be a commutative principal ideal domain, F be the quotient field of k. The canonical map σ : k ↪ F induces the semilinear isomorphism

σ^{n} : k^{n} = \underbrace{k + k + ⋯ + k}_{n} ⟶ F^{n} = \underbrace{F + F + ⋯ + F}_{n}, n ≥ 2.

This one defines the map P (σ^{n}) : P̃(k^{n}) ⟶ P (F^{n}). When k = K⟨x⟩ is the ring of formal power series in x of some field, then P (σ^{n}) is bijective [6], [13], [14].

Example 1. Let n ≥ 3 and define the map α by Fig.1.

[figure: Q_{3}=<0,1,0>; Q_{2}=<x,1,0>; Q_{1}=<1,0,0>=<x,0,0>; P_{3}=<0,1,0>; P_{2}=<x,1,0>⊃<αx,α,0>; P_{1}=<1,0,0>⊃<x,0,0>; l_{1}=P_{1}∪P_{3}; α; α^{−1}; α; α^{−1}; α; α^{−1}; l_{2}=P_{2}∪P_{3}⊂l_{1}; L=Q_{1}∪Q_{2}]

Figure 1

The lines l_{1} and l_{2} are defined over the ring k and the line L over the field F.

The map α^{−1} : P (L) ⟶ P̃(l_{1}) is not a collineation. It is clear that Q_{1} ⊂ Q_{2} + Q_{3}. On the other hand,

l_{1} = P_{1} ∪ P_{2} = P_{1} ∪ P_{3} ⊃ l_{2} = P_{2} ∪ P_{3}
⟹ α^{−1}(Q_{1}) = P_{1} ⊈ α^{−1}(Q_{2}) ∪ α^{−1}(Q_{3}) = P_{2} ∪ P_{3}.

Note that ⟨x, 0, 0⟩ ∉ P̃(k^{n}).

Example 2. Suppose that n = 2 and define the harmonic map α : P̃(l) → P (L) by Fig. 2

[figure: Q_{4}=<0,1>=<0,2>; Q_{2}=<x,−1>; Q_{1}=<x,1>; P_{4}=<0,1>; P_{2}=<x,−1>; P_{1}=<x,1>; l=P_{1}∪P_{2}; α; α^{−1}; α; α^{−1}; α; α^{−1}; l⊂l_{1}=P_{1}∪P_{3}; Q_{3}=<1,0>=<2x,0>; P_{3}=<1,0>; α; α^{−1}; L_{1}=Q_{1}∪Q_{2}]

Figure 2

It is easy to see that α is not harmonic, though it is bijective. Note that the lines l and l_{1} are defined over the ring k and the line L over the field F. It is obvious that α^{−1} is not harmonic because the points P_{1}, P_{2}, P_{3}, P_{4} are not strictly collinear.

For the completion of the picture we shall give an example which shows that for the system of points P̃(X) over the principal ideal domain, von Staudt’s theorem is not true.

Example 3 (C. Bartolone and F. Di Franco [13]). Let k =F < x >, where F is a field, charF ≠ 2. Define the bijective map α : P̃(k^{2}) ⟶ P̃(k^{2}) by the equations

α(k(0, 1)) = k(0, 1), α(k(1, 0)) = k(1, 0) for k(f, g) ∈ P̃(k^{2}),
α(k(f, g)) = \begin{cases}k(f, g) & if deg(f) ≡ deg(g) (mod 2), \\ k(−f, g) & if deg(f) ≢ deg(g) (mod 2). \end{cases}

This map is harmonic on both sides but is not induced by the semilinear isomorphism [13], [14].

Further, k is a non-commutative left principal ideal domain. Let us investigate the map

f : 𝔐(X) ⟶ 𝔐(X_{1}),

which preserves the lattice-theoretical operation of union (∪-preserving map).

Thus, such map is defined with its restriction on the projective space P (X), so it is natural for the beginning to consider the map f : P (X) ⟶ 𝔐(X_{1}). Since for our general maps the images of the points are not always points, it is natural to generalize the definition of the harmonic map.

Definition 3. The map f : P (X) ⟶ 𝔐(X_{1}) will be called harmonic if for each quadruple of harmonic points P_{1}, P_{2}, P_{3}, P_{4} ∈ P (X) and their images f(P_{1}), f(P_{2}), f(P_{3}), f(P_{4}) ∈ 𝔐(X_{1}) there exist y_{1}, y_{2} ∈ X_{1} such that

Q_{1} = k_{1}y_{1} ↪ f(P_{1}),
Q_{2} = k_{1}y_{2} ↪ f(P_{2}),
Q_{3} = k_{1}(y_{1} + y_{2}) ↪ f(P_{3}),
Q_{4} = k_{1}(y_{1} − y_{2}) ↪ f(P_{4}),

i.e., the points Q_{1}, Q_{2}, Q_{3}, Q_{4} are in a harmonic relation.

Let F be a quotient field of k. According to U. Brehm [37], consider the tensor product \overline{X} = F ⊗_{k} X and the canonical map i : X ⟶ F ⊗_{k} X. The module X will be considered as a k-submodule of the F -vector space X. It is obvious that F X = ⟨F X⟩ = \overline{X}.

Suppose as well that F_{1} is some skew field and k_{1} is a subring of F_{1}. Let \overline{X}_{1} be a F_{1}-vector space and X_{1} be a k_{1}-submodule of \overline{X}_{1} such that ⟨F_{1}X_{1}⟩ = \overline{X}_{1}.

Proposition 5. Let f : 𝔐(X) → 𝔐(X_{1}) be ∪-preserving map and μ : \overline{X} ⟶ \overline{X}_{1} be a semilinear isomorphism with respect to the isomorphism σ : F ⟶ F_{1}. If there exists a subring K_{1} ↪ F_{1} for which

σ(k) ⊆ K_{1} ⊆ F_{1}, K_{1}μ(X) ⊆ X_{1}, f(kx) = K_{1}μ(x),

then f is a LIP-collineation.

Proof. Let P_{1} = kx_{1} , P_{2} = kx_{2}, P_{3} = kx_{3}, P_{1} ⊆ P_{2} + P_{3} then we have

x_{1} = mx_{2} + nx_{3} ⟹ μ(x_{1}) = σ(m)μ(x_{2}) + σ(n)μ(x_{3}) ⟹ K_{1}μ(x_{1})
⊆ [K_{1}σ(k)μ(x_{2})] ∪ [K_{1}σ(k)μ(x_{3})] ⊆ [K_{1}μ(x_{2})] ∪ [K_{1}μ(x_{3})]
⟹ f(kx_{1}) ↪ f(kx_{2}) ∪ f(kx_{3}) ⟹ f is a collineation,

so that

0 ≠ F_{1}[f(kx)] ∩ F_{1}[f(ky)] = F_{1}[K_{1}μ(x)] ∩ F_{1}[K_{1}μ(y)]
⟹ μ(x) ∈ F_{1}μ(y) ⟹ x ∈ F y

⟹ f preserves linear independence .

Suppose that f : 𝔐(X) → 𝔐(X_{1}) is an LIP-collineation. Let us observe some general facts concerning collineations and harmonic maps.

(l_{1}) From the linear independence of f we get f(0) = 0. It is also clear that dim F_{1}f(kx) = 1 for all x ∈ X. Indeed, let P be a point, i.e., P = kx and dim F_{1}f(P) = 2, then all submodules of this point are one-dimensional and have non-zero intersections. Since in F_{1}f(P) we can always find two non-incident points, we get a contradiction.

(l_{2}) Let us show that if F x_{1} = F x_{2} for x_{1}, x_{2} ∈ F, then F_{1}f(kx_{1}) = F_{1}f(kX_{2}).

By the condition there exists \overline{s}, \overline{r} ∈ F such that \overline{r}x_{1} = \overline{s}x_{2}. Consequently, we can find s, r ∈ k for which rx_{1} = sx_{2}. So we have

k(sx_{2}) ⊆ k(x_{2}), k(sx_{2}) = k(rx_{1}) ⊆ k(x_{1})
⟹ f(k(sx_{2})) ⊆ f(kx_{1}) ∩ f(kx_{2})
⟹ F_{1}f(kx_{1}) = F_{1}f(kx_{2}).

(l_{3}) Define the map

f_{1} : \overline{X} \ 0 ⟶ 𝔐_{F_{1}}(\overline{X}_{1})

in the following way: for x ∈ \overline{X}, x ≠ 0,

f_{1}(x) = F_{1}f(ky), y ∈ X ∩ (F x \ 0).

For each n ∈ ℕ and arbitrary x, y_{1}, …, y_{n} ∈ \overline{X} \ 0 from (l_{2}) we get

F x ∩ (∪_{i=1}^{n}F y_{i}) = 0 ⟹ f_{1}(x) ∩ (∪_{i=1}^{n}f_{1}(y_{i})) = 0
⟹ if F x ≠ F y, then f_{1}(x) ≠ f_{1}(y).

Since f is a collineation for x, y_{1}, y_{2} ∈ \overline{X} \ 0, we have: if x ∈ F y_{1} + F y_{2}, then f_{1}(x) ⊆ f_{1}(y_{1}) + f_{1}(y_{2}).

(l_{4}) By induction we can prove that if x ∈ F x_{1} + F x_{2} + ⋯ + F x_{m}, then f_{1}(x) ⊆ f_{(}x_{1}) + f_{1}(x_{2}) + ⋯ + f_{1}(x_{m}).

For m = 1, 2 and x ∈ F x_{m} the statement is obvious. Let x ∉ F x_{m}, then there exist y, z ∈ \overline{X} such that

x ∈ F (y + z), y ∈ F x_{1} + ⋯ + F x_{m−1}, z ∈ F x_{m}, y ≠ 0.

Consequently, by the induction hypothesis we get

f_{1}(y) ⊆ f_{1}(x_{1}) + ⋯ + f_{1}(x_{m−1}).

On the other hand,

x ∈ F y + F x_{m} ⟹ f_{1}(x) ⊆ f_{1}(y) + f_{1}(x_{m})
⟹ f_{1}(x) ⊆ f_{1}(x_{1}) + f_{1}(x_{2}) + ⋯ + f_{1}(x_{m−1}) + f_{1}(x_{m}).

(l_{5}) In the sequel we shall often use the following fact:

Proposition 6. Let x and y be linear independent elements of \overline{X} and 0 ≠ z ∈ \overline{X}, z ∈ (F x + F y) \ F y. Then there exists 0 ≠ d ∈ F such that F (x + dy) = F z.

It is obvious that F z = F (ax + by) and d = a^{−1}b. It is also obvious that d has only one representation by F z.

(l_{6}) Let B be a basis of \overline{X} and x_{0} be an arbitrary but fixed element of B. Define

μ̃ : B ⟶ \overline{X}_{1}, F_{1}μ̃(x) = f_{1}(x), x ∈ B.

So we have

x_{0} + x ∈ F x_{0} + F x, x ∈ B \ x_{0} ⟹ f_{1}(x_{0} + x)
⊆ f_{1}(x_{0}) + f_{1}(x) = F_{1}μ̃(x_{0}) + F_{1}μ̃(x).

Taking Proposition 6 into consideration, we conclude that there exists d ∈ F, d ≠ 0 such that

f_{1}(x_{0} + x) = F_{1}(μ̃(x_{0}) + dμ̃(x)).

Definition 4. μ(x) =^{def} dμ̃(x), x_{0} ≠ x ∈ B, μ(x_{0}) =^{def} μ̃(x_{0}).

So for all x ∈ B we have f_{1}(x) = F_{1}μ(x) and f_{1}(x_{0} + x) = F_{1}(μ(x_{0}) + μ(x)), where x ∈ B \ x_{0}. Consequently, from (l_{3}) we conclude that μ(B) ⊆ \overline{X}_{1} is a linear independent set.

(l_{7}) Let a ∈ F, x ∈ B \ x_{0}, x_{0} + ax ∉ F x. Then from Proposition 6 we can conclude that there exists only one element σ(a, x) ∈ F_{1} for which

f_{1}(x_{0} + ax) = F_{1}[μ(x_{0}) + σ(a, x)μ(x)].

Note that the theorem of von Staudt deals with the harmonic maps of the projective line, i.e., it considers the case when dim_{p} X = 1. In this situation B = {x_{0}, x} and σ(a, x) = σ(a).

Generally, as we shall show in [9], σ does not depend on x, i.e., σ(a, x) = σ(a).

So σ is an injective map. From the above we conclude that σ(0) = 0, σ(1) = 1.

3. Harmonic Maps Generated by Semilinear Isomorphisms

Let \frac{1}{2} ∈ k, dim_{p} X = 1 and f : P (X) ⟶ 𝔐_{k_{1}}(X_{1}), be a harmonic map (Definition 3). In the previous paragraph we have defined the maps f_{1}, σ, μ. It is clear that a set-theoretical map f_{1} defined on the elements of X can also be considered as the map determined on P (X). Let us show now that σ is either a isomorphism or an anti-isomorphism. Recall that σ : k ⟶ k_{1} is an anti-homomorphism if σ(x + y) = σ(x) + σ(y), σ(xy) = σ(y)σ(x) for all x, y ∈ k. We cannot use the classical theorem of K. von Staudt because, on the one hand, f_{1} is the map of the projective line over a ring which is not in general a skew field, and on the other hand, f_{1} is not bijective.

Consider the lines l = kx_{0} + kx and L = F_{1}μ(x_{0}) + F_{1}μ(x). On the line l the points

k(x_{0} + ax), k(x_{0} + bx),

k(x_{0} + \frac{a + b}{2}x) = k[2x_{0} + (a + b)x], k[(a − b)x] = k(\frac{a − b}{2}x)

are in a harmonic relation. According to the definition of f we have

[figure: F_{1}(σ(a−b)μ(x)); F_{1}(μ(x_{0})+σ(b)μ(x)); F_{1}(μ(x_{0})+σ(a)μ(x)); (a−b)x; x_{0}+bx; x_{0}+ax; f_{1}; f_{1}; f_{1}; F_{1}(μ(x_{0})+σ( \frac{a+b}{2} )μ(x)); x_{0}+ \frac{a+b}{2} x; f_{1}; X; l]

Figure 3

Figure 3 represents the map of the elements of the k-module X on the projective line L over the skew field F_{1}.

Note that since f is a harmonic map, for the elements x_{1}, x_{2}, x_{3}, x_{4} ∈ X̄ we have that if the points kx_{1}, kx_{2}, kx_{3}, kx_{4} are in a harmonic relation, then the points f_{1}(x_{1}), f_{1}(x_{2}), f_{1}(x_{3}), f_{1}(x_{4}) are also in a harmonic relation. Consequently, the quadruple

F_{1}(μ(x_{0}) + σ(a)μ(x)), F_{1}(μ(x_{0}) + σ(b)μ(x)),

F_{1}(μ(x_{0}) + σ (\frac{a + b}{2}) μ(x)), F_{1}(σ(a − b)μ(x) = F_{1}μ(x)

is harmonic. So, taking Proposition 3 into consideration, we get

[ − σ (\frac{a + b}{2})+σ(b)][ − σ (\frac{a + b}{2})+σ(a)]^{−1} =−1 ⇒ σ (\frac{a + b}{2})−σ(b)

= −σ (\frac{a + b}{2}) + σ(a) ⇒ σ (\frac{a + b}{2}) = \frac{σ(a)}{2} + \frac{σ(b)}{2}.

Suppose that in this equation b = 0; then we get σ (\frac{a}{2}) = \frac{σ(a)}{2}. If now we suppose that b = a, then we have

σ (\frac{2a}{2}) = σ(a) = \frac{σ(2a)}{2} ⇒ σ(2a) = 2σ(a) ⇒ σ(a + b)

= σ (\frac{2(a + b)}{2}) = 2σ (\frac{a + b}{2}) = 2 (\frac{σ(a)}{2} + \frac{σ(b)}{2}) = σ(a) + σ(b).

So σ is an additive isomorphism.

Suppose now that [σ(a)]^{−1} = σ(a^{−1}) for every a ∈ F. Then we have

a = a(1 − a)(1 − a)^{−1} = a(1 − a)^{−1} − a^{2}(1 − a)^{−1}

⇒ a + a^{2}(1 − a)^{−1} = 1 + a(1 − a)^{−1} − 1 ⇒ a^{2}[a^{−1} + (1 − a)^{−1}]

= a[a^{−1} + (1 − a)^{−1}] − 1 ⇒ a^{2} = a − [a^{−1} + (1 − a)^{−1}]^{−1}
⇒ σ(a^{2}) = σ(a) − [σ(a)^{−1} + (1 − σ(a))^{−1}]^{−1} = [σ(a)]^{2},
ab + ba = (a + b)^{2} − a^{2} − b^{2} ⇒ σ(ab) + σ(ba)
= [σ(a + b)]^{2} − [σ(a)]^{2} − [σ(b)]^{2} = [σ(a)]^{2} + [σ(b)]^{2}
+ σ(a)σ(b) + σ(b)σ(a) − [σ(a)]^{2} − [σ(b)]^{2}
⇒ σ(ab) + σ(ba) = σ(a)σ(b) + σ(b)σ(a).

From (l_{6}) it is obvious that μ̃, and consequently μ can be defined in many different ways, i.e., for every α ∈ F one can define μ_{1} = αμ, and μ_{1} also has the same meaning. Consequently σ is defined for fixed x_{0} and for fixed μ(x_{0}) ∈ X̄_{1}. If now we start from x_{1} and μ(x_{1}), then in the same way we can construct τ : F ⟶ F_{1}.

In fact, [τ(a)]^{−1} = σ(a^{−1}). Indeed,

f_{1}(ax_{0} + x_{1}) = F_{1}[τ(a)μ(x_{0}) + μ(x_{1})] = F_{1}[μ(x_{0}) + [τ(a)]^{−1}μ(x_{1})]
∥
f_{1}(x_{0} + a^{−1}x_{1}) = F_{1}[μ(x_{0}) + σ(a^{−1})μ(x_{1})] ⇒ [τ(a)]^{−1} = σ(a^{−1});

Similarly,

[σ(a)]^{−1} = τ(a^{−1}).

So we have to prove that σ(a^{−1}) = [σ(a)]^{−1}. Suppose that 1+a and 1−a are units of k. Then the points

P_{1} = k(x_{0} + ax_{1}), P_{2} = k(ax_{0} + x_{1}),
P_{3} = k[(x_{0} + ax_{1}) + (ax_{0} + x_{1})] = k[(1 + a)(x_{0} + x_{1})] = k(x_{0} + x_{1}),
P_{4} = k[(x_{0} + ax_{1}) − (ax_{0} + x_{1})] = k[(1 − a)x_{0} + (a − 1)x_{1}]
= k[(1 − a)(x_{0} − x_{1})] = k(x_{0} − x_{1})

are in a harmonic relation. On the other hand, consider the points

Q_{1} = k(ax_{0} + a^{2}x_{1}), Q_{2} = k(a^{2}x_{0} + ax_{1}),
Q_{3} = k[(a + a^{2})(x_{0} + x_{1})], Q_{4} = k[(a − a^{2})(x_{0} − x_{1})].

It is obvious that they are in a harmonic relation while they are strictly collinear, i.e., Q_{i} ⊆ Q_{j} ∪ Q_{k}, 1 ≤ i, j, k ≤ 4.

We have

(a + a^{2})(x_{0} + x_{1}) + (−a^{2}x_{0} − ax_{1}) = ax_{0} + a^{2}x_{1} ∈ k(a^{2}x_{0} + ax_{1})

+ k[(a + a^{2})(x_{0} + x_{1})] ⇒ Q_{1} ∈ Q_{2} ∪ Q_{3};
(a − a^{2})(x_{0} − x_{1}) + (a + a^{2})(x_{0} + x_{1}) = 2ax_{0} + 2a^{2}x_{1}
⇒ 2(ax_{0} + a^{2}x_{1}), ax_{0} + a^{2}x_{1} ∈ k[(a + a^{2})(x_{0} + x_{1})]
+ k[(a − a^{2})(x_{0} − x_{1})] ⇒ Q_{1} ∈ Q_{3} ∪ Q_{4}.

All other inclusions can be proved similarly. Further,

f_{1}[a(x_{0} + ax_{1})] = f_{1}(x_{0} + ax_{1}) = f_{1}(ax_{0} + a^{2}x_{1}) =
= F_{1}[μ(x_{0}) + σ(a)μ(x_{1})] = L_{1},
f_{1}(ax_{0} + x_{1}) = f_{1}(a^{2}x_{0} + ax_{1}) = F_{1}[τ(a)μ(x_{0}) + μ(x_{1})]
= F_{1}[μ(x_{0}) + [τ(a)]^{−1}μ(x_{1})] = L_{2},
f_{1}[(a + a^{2})(x_{0} + x_{1})] = f_{1}(x_{0} + x_{1}) = F_{1}[μ(x_{0}) + μ(x_{1})] = L_{3},
f_{1}[(a − a^{2})(x_{0} − x_{1})] = f_{1}(x_{0} − x_{1}) = F_{1}[μ(x_{0}) − μ(x_{1})] = L_{4}.

Either the quadruple P_{1}, P_{2}, P_{3}, P_{4} or the quadruple Q_{1}, Q_{2}, Q_{3}, Q_{4} is harmonic, so we get that the quadruple L_{1}, L_{2}, L_{3}, L_{4} is also harmonic. For the points L_{1}, L_{2}, L_{3}, L_{4} we have

D_{41} = | \begin{matrix}1, & −1 \\ 1, & σ(a) \end{matrix} |, D_{42} = | \begin{matrix}1, & −1 \\ 1, & [τ(a)]^{−1} \end{matrix} |,

D_{32} = | \begin{matrix}1, & 1 \\ 1, & [τ(a)]^{−1} \end{matrix} |, D_{31} = | \begin{matrix}1, & 1 \\ 1, & σ(a) \end{matrix} |.

Let now P̃_{1}, P̃_{2}, P̃_{3}, P̃_{4} be arbitrary strictly collinear points over the ring k. Then we can assume that

P̃_{1} = ke_{1}, P̃_{2} = ke_{2}, P̃_{3} = k(e_{1} + e_{2}),
P̃_{4} = k(e_{1} + se_{2}), [P̃_{1}, P̃_{2}, P̃_{3}, P̃_{4}] = [s].

As 1−t^{−1}st = t^{−1}(1−s)t we can conclude: if s passes through the whole class of conjugate elements, then 1 − s is also the whole class of conjugate elements. Consequently, to each class [s] there corresponds the class [1 − s]. Taking into consideration that

P̃_{4} = k(e_{1} + se_{2}), P̃_{1} = k(−e_{1}), P̃_{2} = [k(e_{1} + se_{2}) + (−e_{1})],
P̃_{3} = k[(e_{1} + se_{2}) + (1 − s)(−e_{1})] = k[s(e_{1} + e_{2})],

we conclude that for arbitrary strictly collinear points the equation

[P̃_{4}, P̃_{1}, P̃_{2}, P̃_{3}] = 1 − [P̃_{1}, P̃_{2}, P̃_{3}, P̃_{4}]

is true.

Turning back to our consideration, we can check

[L_{4}, L_{1}, L_{2}, L_{3}] = 1 − [L_{1}, L_{2}, L_{3}, L_{4}] = 2.

From Proposition 3 we get [P_{1}, P_{2}, P_{3}, P_{4}] = D_{14}D_{24}^{−1}D_{23}D_{13}^{−1}. Redenote L_{4} = L̄_{1}, L_{1} = L̄_{2}, L_{2} = L̄_{3}, L_{3} = L̄_{4}; then we have

2 = [L̄_{1}, L̄_{2}, L̄_{3}, L̄_{4}] = D̄_{14}[D̄_{24}]^{−1}D̄_{23}[D̄_{13}]^{−1}

= |\begin{matrix}1, & −1 \\ 1, & 1 \end{matrix}| · |\begin{matrix}1, & σ(a) \\ 1, & 1 \end{matrix}|^{−1} · |\begin{matrix}1, & σ(a) \\ 1, & σ(a^{−1}) \end{matrix}| · |\begin{matrix}1, & −1 \\ 1, & σ(a^{−1}) \end{matrix}|^{−1}

= 2[1 − σ(a)]^{−1}[σ(a^{−1}) − σ(a)][σ(a^{−1} + 1]^{−1}

⇒ [σ(a^{−1}) − σ(a)][σ(a^{−1}) + 1]^{−1} = [1 − σ(a)]

⇒ σ(a^{−1})−σ(a)=[1−σ(a)][σ(a^{−1})+1]=1+σ(a^{−1})−σ(a)σ(a^{−1})−σ(a)

⇒ σ(a)σ(a^{−1}) = 1 ⇒ σ(a^{−1}) = [σ(a)]^{−1}.

So we have constructed the map σ with the following properties:

(1) σ(0) = 0, σ(1) = 1;

(2) σ is an additive isomorphism;

(3) [σ(a)]^{−1} = σ(a^{−1});

(4) σ(ab) + σ(ba) = σ(a)σ(b) + σ(b)σ(a).

Thus, σ satisfies the conditions of the Theorem 1.15 from [1]. Taking into consideration that this theorem formulated for skew fields is also true for general rings we conclude that σ is either an isomorphism or an antiisomorphism (see, also, [2], [29], [32]–[34]).

Let k_{1} and k_{2} be arbitrary rings. The map σ : k_{1} ⟶ k_{2} will be called a semi-isomorphism if it is either an isomorphism or an anti-isomorphism. So for fixed x_{0} ∈ B and μ : B ⟶ X̄_{1} we can construct a semi-isomorphism σ : F ⟶ F_{1} (see (l_{6})). If now we replace μ by μ_{1} = ᾱμ, this will influence σ.

Definition 5. Let X_{1} and X_{2} be vector spaces over the skew fields k_{1} and k_{2}, dim X_{1} = dim X_{2} = 2 and σ : k_{1} ⟶ k_{2} is an anti-isomorphism. The map μ : X_{1} ⟶ X_{2} will be called a semilinear isomorphism with respect to σ (σ is a semilinear anti-isomorphism), if μ is defined on the basis B (i.e., for e_{1}, e_{2} the images μ(e_{1}) and μ(e_{2}) are fixed), and then we shall continue as follows:

(i) μ(ae_{i}) = σ(a)μ(e_{i}), i = 1, 2;
(ii) μ(a_{1}e_{1} + a_{2}e_{2}) = [σ(a_{2})]^{−1}μ(e_{1}) + [σ(a_{1})]^{−1}μ(e_{2}).

for each a, a_{1}, a_{2} ∈ k, a_{1}, a_{2} ≠ 0.

It is clear that μ_{0} = 0 and μ(e_{1} ± e_{2}) = μ(e_{1}) ± μ(e_{2}).

Now let us turn back to our considerations. We have the following alternatives:

(i) σ is an isomorphism. Then

f_{1}(ax_{0} + a_{1}x_{1}) = f(x_{0} + a_{0}^{−1}a_{1}x_{1}) = F_{1}[μ(x_{0}) + σ(a_{0}^{−1}a_{1})μ(x_{1})]

= f_{1}[μ(x_{0}) + [σ(a_{0})]^{−1}σ(a_{1})μ(x_{0})] = F_{1}[σ(a_{0})μ(x_{0}) + σ(a_{1})μ(x_{1})].

(ii) σ is an anti-isomorphism. Then

f_{1}(a_{0}x_{0} + a_{1}x_{1}) = f_{1}(x_{0} + a_{0}^{−1}a_{1}x_{1}) = F_{1}[μ(x_{0}) + σ(a_{0}^{−1}a_{1})μ(x_{1})]

= F_{1}[σ(a_{1}^{−1}a_{0})μ(x_{0}) + μ(x_{1})] = F_{1}[μ(x_{0}) + σ(a_{1})[σ(a_{0})]^{−1}μ(x_{1})]

= F_{1}[σ(a_{0})[σ(a_{1})]^{−1}μ(x_{0}) + μ(x_{1})] = F_{1}[[σ(a_{1})]^{−1}μ(x_{0}) + [σ(a_{0})]^{−1}μ(x_{1})].

Thus, for fixed x_{0} ∈ B and μ : B ⟶ X̄_{1} we have defined the semiisomorphism σ and the σ-semilinear (anti)-isomorphism μ, though for all x ∈ X̄ it is true that f_{1}(x) = F_{1}μ(x).

Define the subring K_{1} ↪ F_{1} as follows: f(kx_{0}) = K_{1}μ(x_{0}). It is clear that K_{1} is a k_{1}-submodule in F_{1}. Let us show that

(a) f(kx_{1}) = K_{1}μ(x_{1}),
(b) f(k(a_{0}x_{0} + a_{1}x_{1})

= \begin{cases}K_{1}([σ(a_{1})]^{−1}μ(x_{0})+[σ(a_{0})]^{−1}μ(x_{1}) & if σ is an anti-isomorphism, \\ K_{1}(σ(a_{0})μ(x_{0})+σ(a_{1})μ(x_{1}) & if σ is an isomorphism. \end{cases}

(a) The k-points kx_{0}, kx_{1}, k(x_{0} +x_{1}), k(x_{0} −x_{1}) are harmonic and kx_{0} ⊂ k(x_{0} ± x_{1}) + kx_{1}. From this in general we cannot conclude that f(kx_{0}) ⊂ f(k(x_{0} ± x_{1})) + f(kx_{1}), though F_{1}μ(x_{0}) ⊆ F_{1}[μ(x_{0}) ± μ(x_{1})] + F_{1}μ(x_{1}), and the points F_{1}μ(x_{0}), F_{1}μ(x_{1}), F_{1}[μ(x_{0}) + μ(x_{1})], F_{1}[μ(x_{0}) − μ(x_{1})] are harmonic. By the definition of the harmonic map, in the images f(kx_{0}), f(kx_{1}), f(k(x_{0} + x_{1}), f(k(x_{0} − x_{1})), we can find harmonic k_{1}-points

k_{1}[α_{1}μ(x_{0})], k_{1}[α_{2}μ(x_{1})], k_{1}[α_{3}(μ(x_{0})+μ(x_{1}))], k_{1}[α_{4}(μ(x_{0})−μ(x_{1}))].

In fact, we can choose the elements α_{1}, α_{2}, α_{3}, α_{4} in such a way that α_{i} = α, i = 1, 2, 3, 4:

k_{1}[α_{1}μ(x_{0})] ⊂ k_{1}[α_{2}μ(x_{1})] + k_{1}[α_{3}(μ(x_{0}) − μ(x_{1}))] ⇒ α_{1} = α_{3}β = α_{2}, k_{1}[α_{3}(μ(x_{0}) − μ(x_{1}))] ⊂ k_{1}[α_{1}μ(x_{0})] + k_{1}[α_{2}μ(x_{1})] ⇒ α_{3} = ξα_{1} = γα_{2}.

Consequently, β, ξ, γ are invertible elements so that α_{1} = α_{2} = α_{3}. The same version is also true for α_{4}, i.e., α_{i} = α_{4}. It is obvious that we can choose α such that k_{1}[αμ(x_{0})] ⊆ k_{1}μ(x_{0}) ⊆ f(kx_{0}).

Thus we have

k_{1}[αμ(x_{0})] ⊂ k_{1}[α(μ(x_{0}) ± μ(x_{1}))] + k_{1}[αμ(x_{1})]
⋂ |  ^{⋂ |}  ^{⋂ |}
k_{1}μ(x_{0})  k_{1}[μ(x_{0}) ± μ(x_{1})]  k_{1}μ(x_{1})
⋂  ⋂  ⋂
f(kx_{0})  f[k(x_{0} ± x_{1})]  f(kx_{1})
⋂  ⋂  ⋂
F_{1}μ(x_{0}) ⊂ F_{1}[μ(x_{0}) ± μ(x_{1})] + F_{1}μ(x_{1}) .

Let b ∈ K_{1} be an arbitrary element. Then we get

b[αμ(x_{0})] = α_{1}[α(μ(x_{0}) − μ(x_{1}))] + α_{2}[αμ(x_{1})]

⇒ b = α_{1} = α_{2} ⇒ bμ(x_{1}) ∈ f(kx_{1}) ⇒ K_{1}μ(x_{1}) ⊆ f(kx_{1}).

Suppose now that cμ(x_{1}) ∈ f(kx_{1}). Changing the roles to x_{0} and x_{1}, we get cμ(x_{0}) ∈ f(kx_{0}) = K_{1}μ(x_{0}) ⇒ c ∈ K_{1} ⇒ f(kx_{1}) = K_{1}μ(x_{1}). It is clear that f(k(ax_{i})) = K_{1}μ(ax_{i}), i = 0, 1 are true for arbitrary a ∈ K.

(b) Suppose that σ is an anti-isomorphism. The k-points

k(a_{0}x_{0}), k(a_{1}x_{1}), k(a_{0}x_{0} + a_{1}x_{1}), k(a_{0}x_{0} − a_{1}x_{1})

are harmonic. So we have

f(k(a_{i}x_{i})) ↪ F_{1}μ(a_{i}x_{i}), i = 0, 1;
f[k(a_{0}x_{0} ± a_{1}x_{1})] ↪ F_{1}([σ(a_{1})]^{−1}μ(x_{0}) + [σ(a_{0})]^{−1}μ(x_{1})].

Let b ∈ K_{1}, then

bμ(a_{0}x_{0}) = bσ(a_{0})μ(x_{0}) ∈ F_{1}([σ(a_{1})]^{−1}μ(x_{0})
+ [σ(a_{0})]^{−1}μ(x_{1})) + F ([σ(a_{1})]^{−1}μ(x_{0}) − [σ(a_{0})]^{−1}μ(x_{1}))
⇒ bσ(a_{0})μ(x_{0}) = a_{1}([σ(a_{1})]^{−1}μ(x_{0}) + [σ(a_{0})]^{−1}μ(x_{1}))
+ a_{2}([σ(a_{1})]^{−1}μ(x_{0}) − [σ(a_{0})]^{−1}μ(x_{1}))

⇒ a_{1} =a_{2} ⇒bσ(a_{0})μ(x_{0})=2a_{1}[σ(a_{1})]^{−1}μ(x_{0})⇒2a_{1} =bσ(a_{0})σ(a_{1})

⇒ bσ(a_{0})σ(a_{1})([σ(a_{1})]^{−1}μ(x_{0}) + [σ(a_{0})]^{−1}μ(x_{1}))

∈ f[k(a_{1}a_{0}(a_{0}x_{0} + a_{1}x_{1}))] ↪ F_{1}σ(a_{0})σ(a_{1})[[σ(a_{1})]^{−1})μ(x_{0})

+ [σ(a_{0})]^{−1}μ(x_{1})] ⇒ K_{1}([σ(a_{1})]^{−1}μ(x_{0})
+ [σ(a_{0})]^{−1}μ(x_{1})) ⊆ f[k(a_{0}x_{0} + a_{1}x_{1})].

On the other hand, if

c([σ(a_{1})]^{−1}μ(x_{0}) + [σ(a_{0})]^{−1}μ(x_{1})) ∈ f[k(a_{0}x_{0} + a_{1}x_{1})],

then we get

c[σ(a_{1})]^{−1}μ(x_{0}) = cμ(a_{1}^{−1}x_{0}) ∈ f[kμ(a_{1}^{−1}x_{0})]
= K_{1}μ(a_{1}^{−1}x_{0}) = K_{1}[σ(a_{1})]^{−1}μ(x_{0});
c[σ(a_{0})]^{−1}μ(x_{1}) ∈ K_{1}[σ(a_{0})]^{−1}μ(x_{1}) ⇒ c ∈ K
⇒ f[k(a_{0}x_{0}+a_{1}x_{1})]=K_{1}([σ(a_{1})]^{−1}μ(x_{0})+[σ(a_{0})]^{−1}μ(x_{1})).

The case where σ is an isomorphism is easier and can be proved by similar arguments.

If α ∈ k, x ∈ X̄ \ 0, then we have

K_{1}σ(α)μ(x) = K_{1}μ(αx) = f[k(αx)] ⊆ f(kx)
= K_{1}μ(x) ⇒ K_{1}σ(k) ⊆ K_{1}.

In general, the constructed subring and the maps μ and σ are not unique. If 0 ≠ a ∈ F, then K_{2} := K_{1}a^{−1} is a k_{1}-submodule and μ_{1} := aμ is the semilinear (anti)-isomorphism with respect to σ_{1} = aσa^{−1}. In fact

K_{2}σ_{1}(k) = K_{1}a^{−1}aσ(k)a^{−1} = K_{1}σ(k)a^{−1} = K_{1}a^{−1} = K_{2}
⇒ K_{2}μ_{1}(x) = K_{1}a^{−1}aμ(x) = K_{1}μ(x).

Consequently, there exists a ring K_{1} such that 1 ∈ K_{1}. In fact, K_{1} and μ can be constructed up to a constant factor.

Thus the following inclusions are true:

σ(k) ↪ K_{1}, σ(k) ↪ k_{1} ↪ F_{1}.

By the definition of f : 𝔐(x) ⟶ (X_{1}) we have K_{1}μ(X) ⊆ X_{1}. Thus we prove

Theorem 1 (Representation of Harmonic Maps by the Semilinear Isomorphisms). Let k be a non-commutative left principal ideal domain, \frac{1}{2} ∈ k and X be a torsion-free module over k, dim_{p} X = 1. If f : P (X) ⟶ 𝔐_{k_{1}}(X_{1}) is a harmonic map, then there exist a semiisomorphism σ : F ⟶ F_{1}, a σ-semilinear (anti)-isomorphism μ : \overline{X} ⟶ \overline{X}_{1} and a subring K_{1} ↪ F_{1}, 1 ∈ K_{1}, such that K_{1}μ(X) ⊆ X_{1},

[figure: σ(k); K_{1}; k_{1}; F_{1}]

and f(kx) = K_{1}μ(x) for all x ∈ X.

From the theorem we get: if f : P (X) → P (X_{1}) is a bijection, then K_{1} = k_{1} and k_{1}μ(X) = X_{1}. So we have

Corollary (K. von Staudt’s Theorem). Let k be a noncommutative left principal ideal domain,\frac{1}{2} ∈ k; X be a torsion-free module over k, dim_{p} X = 1. The bijection f : P (X) ⟶ P (X_{1}) is harmonic if and only if there exist an isomophism or an anti-isomorphism σ : k ⟶ k_{1} and σsemilinear isomorphism μ : \overline{X} ⟶ \overline{X}_{1} such that f(kx) = k_{1}μ(x) for every x ∈ X.

Proposition 7. Let μ and μ_{1} be the semilinear (anti)-isomorphisms with respect to σ, σ_{1} : F ⟶ F_{1} and dim X̄ ≥ 2. If K and K_{1} (1 ∈ K, K_{1}) are subrings of F such that K_{1}μ_{1}(x) = Kμ(x) for all x ∈ X and

[figure: K; F_{1}; σ(k); K_{1}; k_{1}; F_{1}]

then there exists an element a ∈ K such that

K_{1} = Ka^{−1}, μ_{1} = aμ.

For this suppose that the points F x and F y are distinct. Then there exist a, b, c ∈ F such that

μ_{1}(x) = aμ(x), μ_{1}(y) = bμ(y), μ_{1}(x + y) = cμ(x + y)
⇒ aμ(x) + bμ(y) = μ_{1}(x) + μ_{1}(y) = μ_{1}(x + y)
= cμ(x + y) ⇒ a = b = c ⇒ μ_{1}(z) = aμ(z).

Since F X = \overline{X}, we get μ_{1}(x) = aμ(x) for all x ∈ X̄. Let x ∈ X \ 0; then

Kμ(x) = K_{1}μ_{1}(x) = K_{1}aμ(x) ⇒ K = K_{1}a, K_{1} = Ka^{−1}.

As 1 ∈ K_{1}, it is clear that a ∈ K.

Some of results presented here were announced in [40], [41].

To conclude we would like to note that the remarkable comprehensive monograph [42] has recently been published, describing the state of the art of this field and prospects for further study.

Acknoledgement

The author was partially supported by a grant of DAAD (November 1995–February 1996) and by a Grant of C.N.R. (March 1996–April 1996).

References

1. E. Arthin, Geometric algebra. Interscience, New York, 1957.

2. R. Baer, Linear algebra and projective geometry. Academic Press, New York, 1952.

3. L. Dieudonné, La Géométrie des groupes classiques. Springer-Verlag, Berlin–Heidelberg–New York, 1951.

4. P. Plaumann and K. Strambach, eds.,Geometry von Staudt’s point of view. Reidel Publ. Co., Dordrecht–Boston–London, 1980.

5. R. Kaya, P. Plaumann, K. Strambach, eds., Rings and geometry. Reidel Publ. Co., Dordrecht–Boston–London, 1985.

6. M. Ojanguren and R. Sridharan, A note on the fundamental theorem of projective geometry. Comment. Math. Helv. 44(1969), No. 3, 310–315.

7. G. Ancochea, Le théorème de von Staudt en géométrie projective quaternionienne. J. Reine Angew. Math. 184(1942), 193–198.

8. N. B. Limaye, Projectivities over local rings. Math. Z. 121(1971), 175–180.

9. N. B. Limaye, Cross-ratios and projectivities of line. Math. Z. 129(1972), 43–53.

10. H. Schneffer, Das von Staudtsche Theorem in der Geometrie der Algebren. J. Reine Angew. Math. 267(1974), 133–142.

11. B. R. McDonald, Projectivities over rings with many units. Comm. Algebra 9(1981), 195–204.

12. B. V. Limaye and N. B. Limaye, Fundamental theorem for the projective line over non-commutative local rings. Arch. Math. 28(1977), 102–109.

13. C. Bartolone and F. Di Franco, A remark of the projectivities of the projective line over a commutative ring. Math. Z. 169(1979), 23–29.

14. C. Bartolone and F. Bartolozzi, Topics in geometric algebra over rings. Rings and geometry, NATO Adv. Stud. Inst., 353–389, Reidel, 1985.

15. J. C. Ferrar and F. D. Veldkamp, Neighbor-preserving homomorphisms between projective ring planes. Geom. Dedicata 18(1985), 11–33.

16. J. C. Ferrar and F. D. Veldkamp, Admissible subrings revisited. Geom. Dedicata 23(1987), 229–236.

17. F. D. Veldkamp, Projective ring planes and their homomorphisms. Rings and geomerty, NATO Adv. Stud. Inst., 289–350, Reidel, 1985.

18. F. D. Veldkamp, Distant-preserving homomorphisms between projective ring planes. Indag. Math. 47(1985), 443–453.

19. F. Buekenhout, Une généralization du théorème de von Staudt–Hua. Acad. Roy. Belg. Bull. CI Sci. 51(1965), 1282–1293.

20. D. G. James, Projective homomorphisms and von Staudt’s theorem. Geom. Dedicata 13(1982), 291–294.

21. V. Havel, Harmonical quadruplet in Moufang plane. Czechoslovak Math. J. 5(1965), 76–82.

22. V. Havel, On the theorem of Staudt in Moufang plane. Czechoslovak Math. J. 5(1965), 83–90.

23. J. C. Ferrar, Cross-ratios in projective and affine planes. Geometry – von Staudt’s point of view, P. Plaumann, K. Strambach, eds., 289–350, Reidel, 1985.

24. W. Benz, Zum Büschelsatz in der Geometrie der Algebren. Monatsh. Math. 77(1973), 1–9.

25. W. Benz, Vorlesungen über Geometrie der Algebren. Springer-Verlag, Berlin–Heidelberg–New York, 1973.

26. W. Benz, H.-J. Samaga, and H. Schaeffer, Cross-ratios and a unifying treatment of von Staudt’s notion of reeler Zug. Geometry – von Staudt’s point of view, P. Plaumann, K. Strambach, eds., 126–150, Reidel, 1985.

27. A. Dress and W. Wenzel, On combinatorial and projective geometry. Geom. Dedicata 34(1990), 161–197.

28. D. James, W. Waterhouse, and B. Weisfeiler, Abstract homomorphisms of algebraic groups: problems and bibliography. Comm. Algebra 9(1981), No. 1, 95–114.

29. A. J. Hoffman, A note on cross ratios. Amer. Math. Monthly 58(1951), 613–614.

30. B. N. Limaye and N. B. Limaye, The fundamental theorem for the projective line over commutative rings. Aequationes Math. 16(1977), 275– 281.

31. C. Bartolone, Jordan homomorphisms, chain geometries and the fundamental theorem. Abh. Math. Sem. Univ. Hamburg 59(1989), 93–99.

32. I. N. Herstein, Jordan homomorphisms. Trans. Amer. Math. Soc. 81(1956), 331–351.

33. N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings. Trans. Amer. Math. Soc. 69 (1950),479–502.

34. L. K. Hua, On the automorphisms of field. Proc. Nat. Acad. Sci. U.S.A. 35(1949), 385–389.

35. B. R. McDonald, Geometric algebra over local rings. Marcel Dekker, New York, Basel, 1976.

36. B. McDonald, GL(2) of rings with many units. Comm. Algebra 9(1980), No. 2, 205–220.

37. U. Brehm, Untermodulverbände torsionfreier Modulen. Ph. D. Thesis, Freiburg Universität. Freiburg im Br., 1983.

38. N. Kulkarni, Fundamental theorem of projective geometry over a commutative ring. Indian J. Pure Appl. Math. 11(1980), 1561–1563.

39. A. Lashkhi, Collineations and hemitrophisms over rings. Georgian Math. J. (to appear).

40. A. A. Lashkhi, Harmonic maps and collineations of modules. (Russian) Bull. Acad. Sci. Georgian SSR 133(1989), No. 3, 497–500.

41. A. A. Lashkhi, Harmonic maps of modules. (Russian) Mat. Zametki 47(1990), No. 1, 161–163.

42. Handbook of incidence geometry. Buildings and foundations. Edited by F. Buekenhout, North-Holland, Elsevier, Amsterdam etc., 1995.

(Received 01.09.1993; revised version 22.04.1996)

Author’s address: Department of Applied Mathematics Georgian Technical University 77, Kostava St., Tbilisi 380075 Republic of Georgia