On groupoid graded von Neumann regular rings and a Brandt groupoid graded Leavitt path algebras
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 $x\in xRx$ for every homogeneous element $x\in R.$ 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 $s\in S,$ there exist $s^{-1}\in S$ and idempotent elements $e,$ $f\in S$ for which $es=sf=s,$ $fs^{-1}=s^{-1}e=s^{-1},$ $ss^{-1}=e$ 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 $\mathbb{Z},$ 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 $\mathbb{Z}$-grading of Leavitt path algebras.