On the Jacobson radical of a groupoid graded ring

Abstract

Let R be a ring graded by a finite cancellative partial groupoid S, and let E(S) denote the set of all idempotent elements of S. The Jacobson radical of a ring A is denoted by J(A), and P(A) denotes the prime radical of A. In this paper we affirmatively answer two questions posed by Andrei Kelarev. In particular we prove that: i) J(R)∩Re=J(Re) for every e∈E(S); ii) J(R)n is contained in the largest homogeneous quasi-regular ideal of R for some integer n. We moreover prove that P(R)∩Re=P(Re) for every e∈E(S), and investigate the questions of nilness and nilpotency of the Jacobson radical J(R).