Semigroup Rings and Semilattice Sums of Rings

Abstract

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.

Radicals of supplementary semilattice sums of associative rings

“…It therefore follows from the exact sequence Some information is given by our next result, in the course of the proof of which we show that strict radical properties satisfy Weissglass' condition (F) [11]. PROPOSITION …”

“…Hence R/I is ^-semisimple, so I = Jf g (R). A special case of this corollary is given in [11]. Proposition 3 holds for strict, hereditary radical classes 0t such that the property of 5?-semi-simplicity satisfies condition (F) of [11].…”

“…Semilattice sums of rings were introduced by Weissglass [11]. Let Ω be a semilattice, a commutative semigroup in which all elements are idempotent.…”

“…In [11], Weissglass considered the inheritance of properties by a supplementary semilattice sum R = Σ αeΩ # α from its subrings R a . In [8], Janeski and Weissglass proved that R is regular if and only if each R a is.…”

Regularity of Semilattice Sums of Rings

“…One immediate consequence is a theorem due to Weissglass [7,Theorem 3] that states that if D is a semilattice of groups Gx, a e £2, then RD is regular if and only if RGX is regular for every a eü,. However, a more sophisticated application of Theorem 5, together with a result of Putcha on the structure of semigroups yields a more comprehensive result.…”