Semigroup rings and semilattice sums of rings

Abstract

A generalization of the concept of a decomposition of a ring into a direct sum of ideals is introduced. The question of semisimplicity of the ring in terms of the semisimplicity of its summands is investigated. The results are applied to semigroup rings. Introduction.Many substantial results in the theory of rings either concern themselves with or use a decomposition of the ring into a direct sum of ideals. In this paper we introduce a generalization of direct sum decompositions. A ring is a supplementary semilattice sum of subrings if as an additive abelian group it is a direct sum of the additive groups of the subrings and if the subrings multiply in a certain natural way. We begin the investigation of semilattice sum decompositions by showing that in certain cases the semisimplicity of the subrings implies the semisimplicity of the ring. The investigation is carried out in the context of 7r-semisimplicity where 77 is a hereditary, homomorphic invariant property of rings and thus the result holds for Jacobson semisimplicity, nil semisimplicity and nilpotent semisimplicity (semiprime). The results are then applied to semigroup rings. We prove that if R is a commutative ring with identity and D is a commutative semigroup such that a power of each element lies in a subgroup then the semigroup ring RD is semiprime if and only if D is a semilattice of groups Ga, a e Q, and RGa is semiprime for every a e O. This generalizes Theorem 5.21 of Clifford and Preston [1]. In the paper subsequent to this one Janeski and the author study regularity of semilattice sum decompositions.

On the Index of Nilpotency of Semigroup Graded Rings

“…Strong supplementary semilattice sums of rings have been the subject of study by many authors for various ring theoretic properties since it was introduced by Weissglass [15] in 1973. The notion of a strong band graded ring and more generally a strong semigroup graded ring are generalizations of this concept and have also been studied considerably.…”

On Semisimple Semigroup Rings

“…Then the semigroup ring R[S] consists of all formal sums ESsrSs such that r, E R and r, = 0 for all but finitely many s E S; addition and multiplication are defined in the obvious manner (see [1], [2], [3], [6], [7]). Then the semigroup ring R[S] consists of all formal sums ESsrSs such that r, E R and r, = 0 for all but finitely many s E S; addition and multiplication are defined in the obvious manner (see [1], [2], [3], [6], [7]).…”

Untitled

“…Let A be a ring and let S be a semigroup. Then the semigroup ring R[S] consists of all formal sums ~Zsesrss such that rs E R and rs = 0 for all but finitely many j G S; addition and multiplication are defined in the obvious manner (see [1], [2], [3], [6], [7]). Several authors have studied properties it for specialized semigroup rings.…”

“…In [7], a ring T is called a supplementary semilattice sum of subrings Ta (a E P) if the following conditions hold: T = 2aS7»Ta, TaTß C Tap for all a, ß E P, and Ta n (I>a¥=ßTß) = 0 for each a E P. Clearly, R[S] is always a supplementary semilattice sum of subrings with Ta = R[Sa]. If t E T, we define P-supp t = ( a G P\t = 2 ra and /" ^= OJ.…”

“…Clearly there exists a ß E P satisfying the following conditions, (a) ß E P-snpp x, and (b) if a G P-supp x, then a ^> ß. (In fact, by [7,Lemma 2] there will be more than one such ß.) Let P' = (o G P\a < y and y G P-supp x); then P' is a semilattice.…”