Semigroup graded rings

Abstract

Let S be a semigroup. A ring R with a direct sum decomposition R = ⊕ Rs such that RsRt ⊆ Rst for elements s and t in S is called a semigroup graded ring. -- In this thesis, we develop techniques for studying such rings based on the structure theory of semigroups. We apply these techniques to investigate various ring theoretic properties of semigroup graded rings. -- In many cases, we relate a property of R to the components Re indexed by idempotent elements e of the grading semigroup S. If S is finite, then R is perfect, semilocal, or semiprimary if and only if the same is true of each such component Re. We prove that the nilpotency of the Jacobson radical of each Re is sufficient for the nilpotency of the Jacobson radical of R for rings graded by finite semigroups, and obtain a similar condition for the Jacobson radical to be locally nilpotent for rings graded by locally finite semigroups. We also show that R is a Jacobson ring if each Re is a Jacobson ring and S is finite. -- We show that cancellativity is a necessary condition on a semigroup S in order that the Jacobson radical of each S-graded ring be homogeneous. With certain restrictions on the graded ring, we completely classify commutative semigroups and regular semigroups for which the Jacobson radical of each S-graded ring is homogeneous. -- A result of Zelmanov that only finite semigroups admit right Artinian semigroup algebras is generalised to show that, under certain conditions, a right Artinian semigroup graded ring necessarily has finite support. -- We find necessary and sufficient conditions for rings graded by elementary Rees matrix semigroups to be semisimple Artinian. These rings are one of the essential pieces in the structure theory of graded rings that we develop herein. --Results on nilpotence and perfectness are generalised to semigroup graded rings with finite support.