Applications of epigroups to graded ring theory
By an epigroup the author means a semigroup \(S\) such that a power of every element \(s \in S\) lies in a subgroup of \(S\). The paper is concerned with rings \(R = \bigoplus_{s \in S} R_ s\) graded by an epigroup \(S\), with an emphasis on classical finiteness conditions and on nil properties of such rings. The main results are in the form of two equivalent conditions (i) and (ii), where (i) is a ring property satisfied by all \(S\)-graded rings and condition (ii) gives a description of the semigroup \(S\). Condition (i) exhibits the derivation of the given property of \(R\) from the corresponding property for the group-graded subrings \(R_ G = \bigoplus_{g \in G} R_ g\) for subgroups \(G\) of \(S\). Here is a sample result: for a semigroup \(S\) the following conditions are equivalent: (i) each \(S\)-graded ring \(R\) is nilpotent provided that the subrings \(R_ G\) are nilpotent for all maximal subgroups \(G\) of \(S\), (ii) \(S\) is an epigroup with a finite number of idempotents and every nil factor of \(S\) is nilpotent. The paper contains a list of problems and an extensive bibliography.
scientific article; zbMATH DE number 763799
scientific article