On groupoid graded von Neumann regular rings and a Brandt groupoid graded Leavitt path algebras

Abstract

Let S be a partial groupoid, that is, a set with a partial binary operation. An S-graded ring R is said to be graded von Neumann regular if x ∈ xRx for every homogeneous element x ∈ R. Under the assumption that S is cancellative, we characterize S-graded rings which are graded von Neumann regular. If a ring is S-graded von Neumann regular, and if S is cancellative, then S is such that for every s ∈ S, there exist s −1 ∈ S and idempotent elements e, f ∈ S for which es = sf = s, f s −1 = s −1 e = s −1 , ss −1 = e and s −1 s = f, which is a property enjoyed by Brandt groupoids. We observe a Leavitt path algebra of an arbitrary non-null directed graph over a unital ring as a ring graded by a Brandt groupoid over the additive group of integers Z, and we prove that it is graded von Neumann regular if and only if its coefficient ring is von Neumann regular, thus generalizing the recently obtained result for the canonical Z-grading of Leavitt path algebras.