On the Jacobson and simple radicals of semigroups

Ilić-Georgijević, Emil

doi:10.2298/fil1807577i

Cited by 7 publications

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References 22 publications

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“…The notion of pseudo-unitaryness also served as motivation to introduce the notion of a pseudounitary homogeneous semigroup in [32]. For the study of homogeneous semigroups [29] in case when they are graded by a group or by a small category all of whose morphisms are invertible, we refer the reader to [15,25]. Cayley graphs of homogeneous semigroups are studied in [32,34], inspired by the results which can be found in [42].…”

Section: If Moreover R Is Nearly Epsilon-strongly Graded Thenmentioning

confidence: 99%

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

Ilić-Georgijević¹

2022

Preprint

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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.

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