MATEMATIQKI VESNIK 48 (1996), 83–85

UDK 512.558

originalni nauqni rad research paper

CHARACTERIZATION OF REGULAR SEMIRINGS

P. Mukhopadhyay

Abstract. The main purpose of this paper is to establish some necessary and sufficient conditions for a semiring to be regular, in terms of its k-ideals.

1. Introduction

A semiring is a non-empty set R equipped with two binary operations, called addition, +, and multiplication, denoted by juxtaposition such that R is multiplicatively a semigroup and additively a commutative semigroup and that the multiplication is distributive with respect to the addition both from the left and from the right. An element denoted by 0 is called the zero of R if a + 0 = a and a0 = 0a = 0 for all a ∈ R. A semiring is said to be regular in the sense of von Neumann (cf. [1]) if for every element a ∈ R, there exist some x, y ∈ R such that a + axa = aya. A k-ideal I of a semiring R is an ideal such that, if a ∈ I and x ∈ R and a + x ∈ I then x ∈ I.

Definition 1.1. Let I be a sebsemiring of a semiring R. Then \overline{I} = \{ a ∈ R | a + x ∈ I for some x ∈ I } is called k-closure of I.

It is easy to check that, if I is an ideal of R, then \overline{I} is a k-ideal. In fact, it is the smallest k-ideal containing I and I = \overline{I} if and only if I is a k-ideal. We now state teh following theorem which was proved in [2].

Theorem 1.2. A semiring R is regular if and only if A ∩ B = \overline{AB} holds for every right k-ideal A and left k-ideal B of R.

2. Semiring I(R)

Let R be a semiring and I(R) be the set of all k-ideals of R. In I(R) we define the following operations of “addition” denoted by ⊕ and “multiplication” denoted by ∘; for any I, J ∈ I(R), I ⊕ J = \overline{I + J}, I∘J = \overline{IJ}, where IJ is the set consisting of all finite sums of the form ∑_{i=1}^{n} a_{i}b_{i}, n ∈ N, with a_{i} ∈ I and b_{i} ∈ J. Through a lengthy but routine calculation it can be shown that (I(R), ⊕, ∘) is a semiring. If R be taken as multiplicatively commutative, then so will be I(R).

AMS Subject Classification: 16Y60

We now prove the following theorem.

Theorem 2.1. A multiplicatively commutative semiring with zero, 0 ≠ 1, is regular if and only if I(R), as defined above, is regular.

Proof. Let R be a multiplicatively commutative regular semiring. Then for any two k-ideals I and J we can write from Theorem 1.2 that I ∩ J = \overline{IJ}. Hence I∘J = \overline{IJ} = I ∩ J. To prove that I(R) is regular, let I ∈ I(R), we see that I ⊕I∘R∘I = I∘R∘I holds. In fact, I ⊕I∘R∘I = I ⊕I ∩R∩I = I ⊕I = \overline{I + I} = \overline{I} = I and I∘R∘I = I ∩ R ∩ I = I, proving the regularity of I(R). Conversely, let I(R) be regular. We want to prove that R is so. Let a ∈ R. We consider the principal k-ideal generated by a, i.e. \overline{Ra} (since R is commutative and 1 ∈ R). By regularity of I(R), we get that there exist A, B ∈ I(R) such that

\overline{Ra} ⊕ \overline{Ra}∘A∘\overline{Ra} = \overline{Ra}∘B∘\overline{Ra}.

We see that a ∈ \overline{Ra} ⊕ \overline{Ra}∘A∘\overline{Ra}. Hence a ∈ \overline{Ra}∘B∘\overline{Ra}, i.e. a ∈ \overline{\overline{\overline{Ra} B} \overline{Ra}}.

Now, \overline{\overline{\overline{Ra} B} \overline{Ra}} ⊆ \overline{\overline{Ra} \overline{Ra}} [∵ \overline{\overline{Ra} B} ⊆ \overline{\overline{Ra}} = \overline{Ra}] = \overline{\overline{aR} \overline{Ra}} [by commutativity]. Hence, a ∈ \overline{\overline{aR} \overline{Ra}}, so that a + ∑_{i=1}^{n} x_{i}y_{i} = ∑_{i=1}^{m} p_{i}q_{i}, where x_{i}, p_{i} ∈ \overline{aR} and y_{i}, q_{i} ∈ \overline{Ra} for all i = 1, 2,…, n or m, as the case may be. We have

x_{i} + at_{i} = as_{i}, p_{i} + at′_{i} = as′_{i}, y_{i} + r_{i}a = r′_{i}a  (α)

for some t_{i}, s_{i}, t′_{i}, s′_{i}, r_{i}, r′_{i} ∈ R with i = 1, 2,…, n or m as the case may be. From (α) we derive

x_{i}y_{i} + x_{i}r_{i}a + at_{i}y_{i} + at_{i}r_{i}a = as_{i}(y_{i} + r_{i}a) = as_{i}r′_{i}a

and at_{i}y_{i} + at_{i}r_{i}a = at_{i}r′_{i}a, hence x_{i}y_{i} + as_{i}r_{i}a + at_{i}r′_{i}a = as_{i}r′_{i}a + at_{i}r_{i}a. Thus we have

x_{i}y_{i} + au_{i}a = av_{i}a

for u_{i} = s_{i}r_{i} + t_{i}r′_{i} and v_{i} = s_{i}r′_{i} + t_{i}r_{i}. Similarly we can get p_{i}q_{i} + au′_{i}a = av′_{i}a for some u′_{i}, v′_{i} ∈ R.

Therefore a + axa = aya, for x = ∑_{i=1}^{n} v_{i} + ∑_{i=1}^{m} u′_{i}, y = ∑_{i=1}^{n} u_{i} + ∑_{i=1}^{m} v′_{i}. hence R is regular.

3. Semiprime k-ideal

Definition 3.1. A k-ideal I of a semiring R is said to be semiprime if and only if I = \sqrt{I}, where \sqrt{I} = \{ a ∈ R | a^{n} ∈ I for some positive integer n \}.

Theorem 3.2. A commutative semiring R is regular if and only if every k-ideal of R is semiprime.

Proof. Let R be a regular semiring and I be any k-ideal of R. Since I ⊆ \sqrt{I} always holds, it will be sufficient to prove the reverse inclusion only. Let 0 ≠ a ∈ \sqrt{I}.

Then a^{n} ∈ I, for some positive integer n. Since also a^{n+1} ∈ I, we can assume that n is odd. By regularity of a, there exist x, y ∈ R such that a + axa = aya, i.e. (by commutativity of R)

a + a^{2}x = a^{2}y,  (1)

i.e. a^{2} + a^{3}x = a^{3}y, i.e. a^{2}x + a^{3}x^{2} = a^{3}yx, i.e. a + a^{2}x + a^{3}x^{2} = a + a^{3}yx, i.e. (by (1)) a^{2}y + a^{3}x^{2} = a + a^{3}yx, i.e. a^{2}y + a^{3}xy + a^{3}x^{2} = a + a^{3}xy + a^{3}yx, i.e.

a^{3}y^{2} + a^{3}x^{2} = a + a^{3}(xy + yx).  (2)

On multiplying both sides of the relation (2) by a repeatedly we get

a^{n}(x^{2} + y^{2}) = a^{n}(xy + yx) + a^{n−2}.  (3)

Now, as I is an ideal and a^{n} ∈ I we get a^{n}(x^{2} + y^{2}) ∈ I and a^{n}(xy + yx) ∈ I. Again as I is a k-ideal we get from (3) a^{n−2} ∈ I. Repeating this process enough number of times ultimately we have a ∈ I. Consequently \sqrt{I} ⊆ I so that I = \sqrt{I}.

Going in the other direction, let us now assume that R is a commutative semiring in which every k-ideal is semiprime, i.e. I = \sqrt{I}. Now, given any 0 ≠ a ∈ R, we consider the k-ideal \overline{Ra^{2}}. As we know that a^{3} ∈ \overline{Ra^{2}} and every k-ideal is semiprime, we get

a ∈ \sqrt{\overline{Ra^{2}}} = \overline{Ra^{2}},

i.e. a + xa^{2} = ya^{2} for some x, y ∈ R, i.e. a + axa = aya (by commutativity on R). Hence R is regular.

Acknowledgement. The author is greatful to his guide Dr. M. K. Sen for suggesting this problem. The author expresses his deep gratitude and sincere thanks to the learned referee for his meticulous reading and valuable suggestions which have definitely improved the final form of this paper.

REFERENCES

[1] S. Bourne, The Jacobson radical of a semiring, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 163– 170

[2] M. K. Sen, P. Mukhopadhyay, Von Neumannian regularity in semirings, Kyungpook Math. J. 35 (1995), 249–258

(received 15.01.1996, in revised form 10.08.1996.)

Department of Pure Mathematics, University of Calcutta 35, Ballygunge Circular Road, Calcutta-700019, India