On the Jacobson radical of a groupoid graded ring

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) boolean AND R-e = J(R-e) for every e is an element of 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) boolean AND R-e. = P(R-e) for every e is an element of E(S), and investigate the questions of nilness and nilpotency of the Jacobson radical J(R). (C) 2021 Elsevier Inc. All rights reserved.