Identities of Regular Semigroup Rings

The author proves that, if S is an FIC-semigroup or a completely regular semigroup, and if RS is a ring with identity, then R < E(S) > is a ring with identity. Throughout this paper, R denotes as a ring with identity. Let S be a semigroup, X ⊆ S . The following notations are used in the paper: < X > : the subsemigroup of S generated by X ; |X| : the cardinal number of X ; E(S): the set of idempotents of S ; RS : the contracted semigroup ring of S over R ; supp(A) = {s ∈ S : rs 6= 0} for 0 6= A = ∑ rss ∈ RS with s ∈ S and rs ∈ R ; IRS : the identity (if it exists) of RS . The following Problem is raised in [1]: Problem 1 Which semigroup rings are rings with identity? It is known that for a semigroup S , the semigroup ring R[S] possesses an identity iff both RS and R do; if RS is a ring with identity, then so is R (see[1]). So it is enough to discuss the existence of identity of RS only. Li Fang investigated the existence of identity of orthodox semigroup rings. The following problem is raised in [2]: Problem 2 Let S be a regular semigroup. If RS is a ring with identity, is R < E(S) > a ring with identity? In the following we shall use the terminology, notation and basic results of [3]. Definition 1. A regular semigroup S is called an FIC-semigroup, if for any sequence {ei}i=1 of E(S), e1 ≥ e2 ≥ e3 ≥ ... ≥ en ≥ · · · , there exists N , such that eN = eN+1 = · · · = eN+m = · · · . Lemma 2. Let S = M [0](G; I,Λ;P ) be a completely [0-] simple semigroup, 0 6= e ∈ E(S). If ea 6= 0, then there exists f ∈ E(S) such that a = fea. Proof. Let e = (p−1 λi , i, λ), a = (g, j, μ). By ea 6= 0, pλj 6= 0, thus (p−1 λj , j, λ)ea= (p−1 λj , j, λ)(p −1 λi pλjg, i, μ) = (p −1 λj pλip −1 λi pλj, j, μ)= (g, j, μ) = a . Lemma 3. ([3]) A [0-] simple semigroup is completely [0-] simple iff it contains a primitive idempotent. Lemma 4. ([1]) Let S be a semigroup, let I be the identity of RS , and let s ∈ S . Then there exist e, f ∈ E(S) ∩ supp(I), such that es = sf = s. ∗ * The author wishes to thank Professor Y. Q. Guo for his help. The author is very grateful to Professor J. S. Ponizovskii for his many useful suggestions.