Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 385-393.

σ-Semisimple Rings

M. Boulagouaz L. Oukhtite

Département de Mathématiques, Faculté des Sciences et Techniques Saïss-Fès B.P. 2202 Route d’Imouzzar, Fès, Maroc

Abstract. The aim of this paper is to give a complete description of σ-rings. Indeed, we define and study a more general class of rings with involution that we call σ-semisimple rings. In particular, we prove that for the left artinian rings with involution, this new definition coincides with the classical definition of semisimple rings.

An involution on a ring A is a map σ : A ⟶ A subject to the following conditions: σ(x + y) = σ(x) + σ(y), σ(xy) = σ(y)σ(x) and σ^{2}(x) = x, for each x, y ∈ A. The most common example of involution is the transpose when we consider the matrix algebra M_{n}(K) over an arbitrary field K. Rings and algebras with involutions have been the object of many studies since von Neumann remarked the role played by the classical adjoint in the algebra of linear operators on a Hilbert space. Especially, the theory of rings with involution has been developped to investigate Lie algebras, Jordan algebras and rings of operators. It was known, that there is a connection between semisimple algebras with involution and the classical semisimple Lie groups (see [6]). Recently, the book of involutions that appeared in 1998, gives more complete description of the new investigations concerning this topic (see [3]). Let A be a ring with unity 1 and let σ be an involution on A. For clarity, it is interesting to elucidate some of the terminology to be used in the sequel. Given a subset B of A, σ(B) will stand for the subset of all involutive images of elements of B. An ideal I of A is called a σ-ideal if σ(I) ⊆ I. Moreover, I is said to be a σ-minimal (resp. σ-maximal) ideal of A if I is minimal (resp. maximal) in the set of nonzero (resp. proper) σ-ideals of A. Observe that if I is an ideal of A, then I + σ(I), Iσ(I), σ(I)I and I ∩ σ(I) are σ-ideals of A. Moreover, if we denote by σ̄ the map from A/I to A/I defined by σ̄(a + I) = σ(a) + I, then σ̄ is a well-defined involution on A/I. Throughout this paper, if (A, σ) and (B, τ) are rings with involutions, we use the notation

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

386 M. Boulagouaz; L. Oukhtite: σ-Semisimple Rings

(A, σ) ≃ (B, τ) to express the existence of a ring isomorphism f : A ⟶ B such that f ∘ σ = τ ∘ f.

1. σ-minimal ideals

Throughout this section, A is a ring with unity and σ is an involution of A. If I and J are ideals of A, we will denote the set of all left A-module homomorphisms from I to J by Hom_{A}(I , J ). Then, f ∈ Hom_{A}(I , J ) is said to be a σ-homomorphism if f ∘ σ = σ ∘ f. We will write Hom_{A}^{σ}(I , J ) for the set of all σ-homomorphisms from I to J. In what follows, for a σ-ideal I of A, we denote by S_{σ}(I) the set of all σ-ideals J of A such that J ⊆ I. Hence, for a nonzero σ-ideal I of A we have S_{σ}(I) = I if and only if I is σ-minimal.

Lemma 1. Let I and J be σ-ideals of A. Suppose that f : I → J is a nonzero σhomomorphism.

1) If I is σ-minimal, then f is injective.

2) If J is σ-minimal, then f is surjective.

Proof. 1) Let x be an element of Ker(f ) and let a ∈ A. Since f is a left module homomorphism, then ax ∈ Ker(f ). Moreover,

f(xa) = f ∘σ(σ(a)σ(x)) = σ ∘f(σ(a)σ(x)) = σ[σ(a)f(σ(x))] = σ ∘f ∘σ(x)σ^{2}(a) = f(x)a = 0.

Consequently, xa ∈ Ker(f ). Which proves that Ker(f ) is an ideal of A. The fact that f ∘ σ(x) = σ ∘ f(x) = 0 yields σ(x) ∈ Ker(f ). Thus σ(Ker(f )) ⊂ Ker(f ). Hence Ker(f ) ∈ S_{σ}(I ). As f ≠ 0, the σ-minimality of I implies that Ker(f ) = 0 and therefore f is injective.

2) Similarly, Im(f ) ∈ S_{σ}(J ). In view of the σ-minimality of J, the fact that f ≠ 0 implies that Im(f ) = J , proving the surjectivity of f. □

Corollary 1. If I is a σ-minimal ideal of A, then End_{A}^{σ}(I ) is a division ring.

To prove the converse of Corollary 1, we need to introduce a new class of σ-ideals. A σideal I of A is said σ-indecomposable if I cannot be written as a direct sum of nonzero σ-ideals: if I = P ⊕ Q, then P = (0) or Q = (0). Note that every σ-minimal ideal of A is σ-indecomposable.

Proposition 1. Let I be a σ-ideal of A such that I = ⊕_{i∈S}I_{i}, where each I_{i} is a σ-minimal ideal of A. Then the following conditions are equivalent:

1) I is a σ-minimal ideal.

2) End_{A}^{σ}(I ) is a division ring.

3) I is a σ-indecomposable ideal.

Proof. 1) ⇒ 2) This follows from Corollary 1.

2) ⇒ 3) Suppose that I = P ⊕ Q, where both P and Q are σ-ideals of A. Let π denote the projection of I on P associated with this decomposition. It is straightforward to check that π ∈ End_{A}^{σ}(I ). Since π(π − id_{I}) = 0, the assumption that End_{A}^{σ}(I ) is a division ring implies that P = (0) or Q = (0).

3) ⇒ 1) This is obvious. □

2. σ-semisimple rings

We introduce now a new class of rings with involution. We say that a ring with involution (A, σ) is σ-semisimple if A is a sum of σ-minimal ideals.

Lemma 2. Let A be a σ-semisimple ring such that A = ∑_{i∈S} I_{i}, where each I_{i} is a σminimal ideal of A. If P is a σ-minimal ideal of A, then there is a subset T of S such that A = P ⊕ (⊕_{j∈T} I_{j}).

Proof. Since I_{i} are σ-minimal and P ≠ A, then there exists some i ∈ S such that I_{i} + P is a direct sum. Indeed, otherwise I_{i} ∩ P = I_{i} for all i ∈ S, which implies that P = A. Applying Zorn’s lemma, there is a subset T of S such that the collection {I_{i} : i ∈ T } ∪ {P } is maximal with respect to independence: (⊕_{i∈T} I_{i}) + P = (⊕_{i∈T} I_{i}) ⊕ P. Setting B = (⊕_{i∈T} I_{i}) + P, the maximality of T implies that I_{i} ∩ B ≠ (0) for all i ∈ S. Then, the σ-minimality of I_{i} yields that I_{i} ∩ B = I_{i}, hence I_{i} ⊆ B for all i ∈ S. Consequently, B = A. □

Corollary 2. For a ring with involution (A, σ), the following conditions are equivalent:

1) A is σ-semisimple.

2) A is a direct sum of σ-minimal ideals.

Example. Let A_{4} be the alternating group on 4 letters. Consider the group algebra ℝ[A_{4}] provided with its canonical involution σ defined by σ(∑_{g∈A_{4}} r_{g}g) = ∑_{g∈A_{4}} r_{g}g^{−1}. From [2], the decomposition of the semisimple algebra ℝ[A_{4}] into a direct sum of simple components is as follows: ℝ[A_{4}] = B_{1} ⊕ B_{2} ⊕ B_{3}, where each B_{i} is invariant under σ. More explicitely, B_{1} ≃ ℝ, B_{2} ≃ ℂ and B_{3} ≃ M_{3}(ℝ). In particular, each B_{i} is a σ-minimal ideal of ℝ[A_{4}]. Consequently, ℝ[A_{4}] is a σ-semisimple ring.

Now, let A be a σ-semisimple ring. Since A is finitely generated (indeed, 1 generates A), then A has finite lenght. Thus A = ⊕_{i=1}^{l}I_{i}, where each I_{i} is a σ-minimal ideal of A. It is easy to verify that each I_{i} is generated by a central symmetric idempotent element e_{i} ∈ A (i.e. e_{i}^{2} = e_{i} and σ(e_{i}) = e_{i}), where 1 = ∑_{i=1}^{l} e_{i}. Moreover, e_{i}e_{j} = 0 for all i ≠ j. In what follows, we denote by S the set of central symmetric orthogonal idempotents of A, i.e. S = {e_{1}, …, e_{l}} such that I_{i} = Ae_{i}.

We say that a central symmetric idempotent e ∈ A is a σ-primitive idempotent of A if e cannot be written as a sum of two orthogonal central symmetric idempotents of A.

Proposition 2. For a σ-semisimple ring A, the following statements hold:

1) For each e_{i} ∈ S, e_{i} is a σ-primitive idempotent.

2) I_{1}, …, I_{l} are the only σ-minimal ideals of A.

3) Every nonzero σ-ideal of A is a direct sum of σ-minimal ideals of A.

4) Every σ-ideal of A is generated by a central symmetric idempotent.

Proof. 1) Suppose e_{i} = f_{1}+f_{2}, where f_{1} and f_{2} are orthogonal central symmetric idempotents. As f_{1}f_{2} = 0 then I_{i} = Ae_{i} = Af_{1}⊕Af_{2}. Since Af_{1} and Af_{2} are σ-ideals of A, the σ-minimality of I_{i} yields that Af_{1} = (0) or Af_{2} = (0). Hence f_{1} = 0 or f_{2} = 0.

2) Let T be a σ-minimal ideal of A. For all 1 ≤ i ≤ l, it is clear that T I_{i} = T e_{i} = e_{i}T = e_{i}AT = I_{i}T. Thus T I_{i} is a σ-ideal of A contained in I_{i}. The fact that I_{i} is σ-minimal, implies that either T I_{i} = (0) or T I_{i} = I_{i}. If T I_{i} = (0) for all 1 ≤ i ≤ l then T A = T = (0) which is impossible. Consequently, there exists some 1 ≤ j ≤ l such that T I_{j} = I_{j}. As T I_{j} is a nonzero σ-ideal contained in T, the σ-minimality of T yields T I_{j} = T. Therefore, T = I_{j}.

3) Let I be a nonzero σ-ideal of A. As II_{i} = I_{i}I is a σ-ideal of A contained in I_{i}, for all 1 ≤ i ≤ l, then either II_{i} = (0) or II_{i} = I_{i}. Since I ≠ (0), the fact that I = IA assures the existence of some 1 ≤ t ≤ l such that I = I_{1} ⊕ ⋯ ⊕ I_{t} (one can arrange the indices to have this equality).

4) Let I be a nonzero σ-ideal of A. From 3), there exists some 1 ≤ t ≤ l such that I = I_{1} ⊕ ⋯ ⊕ I_{t}. Setting e = e_{1} + ⋯ + e_{t}, it is clear that e is a central idempotent element generating I. As σ(e_{i}) = e_{i} for all 1 ≤ i ≤ t, it follows that e is symmetric i.e. σ(e) = e. □

Remark. From Proposition 2, it follows that the decomposition of a σ-semisimple ring into a direct sum of σ-minimal ideals is unique.

Now, recall that a ring with involution (A, σ) is said to be a σ-simple ring if (0) and A are the only σ-ideals of A. Let A = ⊕_{i=1}^{l}I_{i} be a σ-semisimple ring, we have already seen that each I_{i} is generated by a central symmetric idempotent e_{i} such that 1 = ∑_{i=1}^{l} e_{i}. Hence, I_{i} is a subring of A with unity e_{i}. Moreover, I_{i} is a σ-simple ring for all 1 ≤ i ≤ l. Consequently, every σ-semisimple ring is a direct sum of σ-simple rings.

Theorem 1. Let (A, σ) be a ring with involution. The following conditions are equivalent:

1) A is σ-semisimple.

2) Every σ-ideal of A is generated by a unique central symmetric idempotent.

3) Every σ-ideal I of A has a complement in S_{σ}(A), i.e. there is a σ-ideal J such that A = I ⊕ J.

Proof. 1) ⇒ 2) Let I be a σ-ideal of A. The existence of a central symmetric idempotent e generating I, follows from Proposition 2. To prove the uniqueness of e, suppose f a central symmetric idempotent such that I = Af. Then f = fe since f ∈ Ae. Samely, e ∈ Af then e = ef. Since both e and f are central, these two equalities yield e = f.

2) ⇒ 3) Let I be a σ-ideal of A, by hypotheses there exists a central symmetric idempotent e generating I, i.e. I = Ae. Set J = A(1 − e), it is clear that J is a σ-ideal of A such that I ⊕ J = A.

3) ⇒ 1) If A is a σ-simple ring then A is σ-semi-simple. Now, suppose that A is not a σ-simple ring and let I be a nonzero proper σ-ideal of A. Consider the set F = {J | J proper σ-ideal of A, I ⊆ J}. Since F is a non-empty set, then Zorn’s lemma assures the existence of a maximal element L of F. It is clear that L is a σ-maximal ideal of A. As L has a complement in S_{σ}(A), then there exists a σ-ideal N of A such that N ⊕ L = A. Hence (N, σ) ≃ (A/L, σ̄), which proves that N is a σ-minimal ideal of A. Consider P := ∑_{N∈M} N, where M is the set of all σ-minimal ideals of A. As P is a σ-ideal of A, then P ⊕ Q = A for some σ-ideal Q of A. Suppose Q ≠ (0) and write 1_{A} = e_{1} + e_{2}, where e_{1} ∈ P and e_{2} ∈ Q (e_{1} and e_{2} being central symmetric elements), it is easy to verify that (Q, σ) is a σ-ring with unity e_{2}. Q cannot be a σ-simple ring. On the other hand, it is clear that every σ-ideal of Q admits a complement. Reasoning as above, we conclude that Q has a σ-minimal ideal which is also a σ-minimal ideal of A. But this again contradicts the fact that P ∩ Q = (0). Hence Q = (0) and therefore A = P. Consequently, A is a σ-semisimple ring. □

Corollary 3. Let f : (A, σ) ⟶ (B, τ) be a surjective homomorphism of rings with involution. If A is σ-semisimple then B is τ-semisimple.

Proof. From Theorem 1, it suffices to show that every τ-ideal of B is generated by a central symmetric idempotent. Let J be a τ-ideal of B. It is straightforward to check that I = f^{−1}(J) is a σ-ideal of A. The σ-semisimplicity of A implies that I = Ae for some central idempotent e ∈ A. Let μ = f(e), then μ is a central idempotent of B such that J = Bμ. Moreover, τ(μ) = τ ∘ f(e) = f ∘ σ(e) = μ, which ends our proof. □

Corollary 4. If I is a nonzero σ-ideal of a σ-semisimlpe ring A, then (\frac{A}{I}, σ̄) is a σ̄-semisimp-le ring.

Proposition 3. Let e be a central symmetric idempotent of a σ-semisimple ring A. Then the following conditions are equivalent :

1) Ae is a σ-minimal ideal.

2) e is a σ-primitive idempotent.

3) Ae^{+} := {x ∈ Z(Ae) | σ(x) = x} is a field, where Z(Ae) denotes the center of the subring Ae of A.

Proof. 1) ⇒ 2) is clear.

2) ⇒ 1) Since σ(e) = e, then Ae is a σ-ideal of A. Writing A = ⊕_{i=1}^{l}I_{i}, from Proposition 2 there exists some 1 ≤ r ≤ l such that Ae = I_{1} ⊕ ⋯ ⊕ I_{r}. Consequently, e = ee_{1} + ⋯ + ee_{r} = ee_{1} +(ee_{2} +⋯+ee_{r}). As ee_{1} and ee_{2} +⋯+ee_{r} are orthogonal central symmetric idempotents of A, the σ-primitivity of e yields that e = ee_{i} for some unique 1 ≤ i ≤ r. Hence Ae is a nonzero σ-ideal of A contained in I_{i}. Accordingly, Ae = I_{i}, since I_{i} is σ-minimal.

3) ⇒ 1) Writing 1 = ∑_{i=1}^{l} e_{i}, it follows that e = ee_{1} + ⋯ + ee_{l}. Since ee_{i} is an idempotent element of the field Ae^{+}, we then deduce that for 1 ≤ i ≤ l, we have either ee_{i} = 0 or ee_{i} = e. As e ≠ 0, there exists necessarily a unique 1 ≤ i ≤ l such that ee_{i} = e. This implies that Ae ⊆ Ae_{i} = I_{i}. The σ-minimality of I_{i} implies that Ae = I_{i}.

1) ⇒ 3) Assume Ae is a σ-minimal ideal of A. According to Corollary 1, End_{A}^{σ}(Ae) is a division ring. Let f ∈ End_{A}^{σ}(Ae), the fact that σ(f(e)) = f ∘ σ(e) = f(e) implies that f(e) is a symmetric element of Ae. If a is any element of A, then

f(ea) = f ∘ σ(σ(a)e) = σ ∘ f(σ(a)e) = σ[σ(a)f(e)] = σ ∘ f(e)a = f(e)a

Thus f(ea) = f(e)a for all a ∈ A. Since

aef(e) = af(e) = f(ae) = f(ea) = f(e)a = f(e)ea = f(e)ae

it follows that f(e) is a central element of Ae and therefore f(e) ∈ Ae^{+}. Consequently, the map Ψ : End_{A}^{σ}(Ae) ⟶ Ae^{+} defined by Ψ(f) = f(e) is a well-defined injective map. Moreover, if f, g ∈ End_{A}^{σ}(Ae) then

Ψ(f ∘ g) = f(g(e)) = f(eg(e)) = f(e)g(e) = Ψ(f)Ψ(g).

To prove the surjectivity of Ψ, let ae be any element of Ae^{+}. Define g ∈ End_{A}(Ae) by g(e) = ae. On one hand g ∘ σ(be) = g(σ(b)e) = σ(b)ae, for all b ∈ A. On the other hand, since ae is central and e is the unit element of Ae then

σ ∘ g(be) = σ(bae) = aeσ(b) = aeσ(b)e = σ(b)eae = σ(b)ae

hence σ ∘ g = g ∘ σ. This proves that Ψ is a ring isomorphism. □

Remark. It follows from Proposition 3 and the fact that A has only a finite number of σ-minimal ideals that A has a finite number of σ-primitive idempotents, namely e_{1}, …, e_{l}.

Recall that the σ-Socle Soc_{σ}(A) of A is defined to be the sum of all σ-minimal ideals of A. It is clear that Soc_{σ}(A) is a σ-ideal of A. Now, using Soc_{σ}(A), we give a σ-semisimplicity criterion for a ring with involution as follows.

Proposition 4. The following conditions are equivalent:

1) A is σ-semisimple.

2) Soc_{σ}(A) = A.

Proof. Suppose that A is σ-semisimple. Writing A = ⊕_{i=1}^{l}I_{i} where each I_{i} is a σ-minimal ideal of A. It follows from 2) of Proposition 2, that I_{1}, …, I_{l} are the only σ-minimal ideals of A. Consequently Soc_{σ}(A) = A. The converse is immediate by the definition of a σ-semisimple ring. □

We will say that a ring A with involution σ is a σ-artinian ring if S_{σ}(A) satisfies the descending condition. That is, there are no infinite decreasing sequences of elements of S_{σ}(A). Equivalently, A is σ-artinian if every nonempty subset of S_{σ}(A) contains a minimal element.

Remark. It is straightforward to verify that every σ-semisimple ring is σ-artinian.

Let M_{σ} denote the set of all σ-maximal ideals of A, and set:

Rad_{σ}(A)=\{ \begin{matrix}A if M_{σ} = ∅ \\ ∩_{L∈M_{σ}}L otherwise \end{matrix}

Proposition 5. The following statements are equivalent:

1) A is σ-semisimple.

2) A is σ-artinian and Rad_{σ}(A) = (0 ).

Proof. 1) ⇒ 2) Writing A = ⊕_{i=1}^{l}I_{i} where each I_{i} is a σ-minimal ideal of A and setting L_{i} = ⊕_{j≠i}I_{j}, then plainly L_{i} is a σ-maximal ideal of A. Since Rad_{σ}(A) ⊂ ∩_{i=1}^{l} L_{i} and ∩_{i=1}^{l}L_{i} = (0), then Rad_{σ}(A) = (0 ).

2) ⇒ 1) Since Rad_{σ}(A) = (0 ), then M_{σ} is a non-empty set. Let us consider P = {L_{i_{1}} ∩ ⋯ ∩ L_{i_{r}}, where r ∈ ℕ and L_{i_{j}} ∈ M_{σ}}. The fact that A is σ-artinian implies that P has a minimal element, say L_{1} ∩ ⋯ ∩ L_{r} and denote it by I. We claim that I = (0). Indeed, otherwise there exists some L_{j} ∈ M_{σ} such that I ∩ L_{j} ⊂ I and I ∩ L_{j} ≠ I, and this contradicts the minimality of I. Since (A, σ) ≃ (∏_{i=1}^{r} A/L_{i}, σ̄), the σ̄-simplicity of A/L_{i} implies that A is a σ-semisimple ring. □

Since a σ-semisimple ring is a direct sum of σ-simple subrings, it is worthwhile to give some properties of σ-simple rings. For this, observe that every simple ring with involution (A, σ) is a σ-simple ring. The following counterexample shows that the converse is not true.

Counterexample. Let B be a simple ring. We denote by B^{o} the opposite ring of B and by σ the exchange involution defined on A = B ⊕ B^{o} by σ(x, y) = (y, x). It is clear that the ring A is not simple, since the ideals of A are (0), A, {0} × B^{o} and B × {0}. But A is σ-simple. Indeed, the only σ-ideals of A are 0 and A.

Now we give a sufficient condition for a σ-simple ring to be simple. For this, we use the following terminology: we say that σ is anisotropic if

σ(a)a = 0 ⇒ a = 0 : for all a ∈ A.

Proposition 6. Let (A, σ) be a σ-simple ring. If the involution σ is anisotropic, then A is a simple ring.

Proof. Let I be an ideal of A. Using the fact that I ∩ σ(I) is a σ-ideal of A, it follows that either I ∩ σ(I) = (0) or I ∩ σ(I) = A. If I ∩ σ(I) = (0), then σ(x)x = 0 for all x ∈ I. As σ is anisotropic, we then deduce that x = 0, and therefore I = (0). If I ∩ σ(I) = A, then I = A. Consequently, A is a simple ring. □

Proposition 7. Let A be a σ-simple ring and let u be an invertible element of A. If σ(u) = λu for some element λ ∈ Z(A) satisfying σ(λ)λ = 1, then A is int(u) ∘ σ-simple.

Proof. Let τ = int(u) ∘ σ. It is readily verified that τ is a well-defined involution on A. For every ideal I of A, it is easy to show that I is a τ-ideal if and only if I is a σ-ideal, which completes the proof. □

Proposition 8. Let A be a σ-simple ring which is not simple. Then there exists a simple subring B of A such that A = B ⊕ σ(B).

Proof. Let I be a nonzero proper ideal of A. Since I ∩ σ(I) is a σ-ideal of A, then necessarily I ∩ σ(I) = (0). The fact that I + σ(I) is a σ-ideal of A, yields that I + σ(I) = A. Indeed, otherwise I = σ(I) and the σ-semisimplicity of A implies that either I = (0) or I = A, which contradicts our assumption. Accordingly, I ⊕ σ(I) = A. Let J be a nonzero ideal of A that is contained in I. A similar reasoning gives J ⊕ σ(J) = A. Choose any element i ∈ I, then there exist j, j′ ∈ J such that i = j + σ(j′). Hence i − j = σ(j′) ∈ I ∩ σ(I), proving i = j. Therefore, I = J. Consequently, I is a minimal ideal of A. Moreover, it is readily verified that there is an idempotent element e ∈ A satisfying e + σ(e) = 1 such that I = Ae. Hence, I is a simple subring, with unity, of A, proving our proposition. □

Remark. It follows from Proposition 8 that if A is a σ-simple ring which is not simple, then there is an idempotent element e ∈ A satisfying 1 = e + σ(e).

Proposition 9. Let (A, σ) be a semisimple ring with involution. Then A is σ-semisimple.

Proof. According to Theorem 1, it suffices to show that every σ-ideal of A is generated by a central symmetric idempotent. Let I be a σ-ideal of A, the semi-simplicity of A assures the existence of a central idempotent e ∈ A generating I, i.e. I = Ae. Since σ(I) = I, it follows that σ(e) = e, proving our proposition. □

As shown in the following counterexample, the converse of Proposition 9 is not true.

Counterexample. Let B be a simple ring which is not semisimple, such a ring exists for it suffices to choose B not left artinian. Consider the ring A = B × B^{o} provided with the exchange involution σ defined by σ(x, y) = (y, x). We have already seen that A is a σ-semisimple ring. Since B is not semisimple, then necessarily A is not semisimple too.

In the following proposition, we show that for the category of left artinian rings the notions of semisimplicity and σ-semisimplicity are the same.

Proposition 10. Let A be a left artinian ring and let σ be an involution of A. Then the following conditions are equivalent:

1) A is semisimple.

2) A is σ-semisimple.

Proof. 1) ⇒ 2) immediate from Proposition 9.

2) ⇒ 1) Write A = ⊕_{i=1}^{l}B_{i}, where B_{i} is a σ-simple subring of A. Since A is left artinian, then B_{i} is left artinian too, for all 1 ≤ i ≤ l. According to Proposition 8, we have to distinguish to cases :

i) B_{i} is a simple ring. The fact that B_{i} is left artinian implies that B_{i} is semisimple. ii) B_{i} = C_{i} ⊕ σ(C_{i}) for some simple subring C_{i} of B_{i}. Since C_{i} is left artinian, then C_{i} is a semisimple ring. Accordingly, B_{i} is a semisimple ring too. As a finite direct sum of semisimple rings is a semisimple ring, we then deduce that also A is a semisimple ring. □

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M. Boulagouaz; L. Oukhtite: σ-Semisimple Rings 393

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Received July 27, 2000