STRONGLY CLEAN ELEMENTS OF A SKEW MONOID RING

Year 2017, Volume: 21 Issue: 21, 164 - 179, 17.01.2017

A. Karimi Mansoub A Moussavi M Habibi

https://doi.org/10.24330/ieja.296324

Abstract

Let R be an associative ring with an endomorphism σ and F ∪ {0}
the free monoid generated by U = {u1, . . . , ut} with 0 added, and M a factor
of F setting certain monomial in U to 0, enough so that, for some n, Mn = 0.
Then we can form the skew monoid ring R[M; σ]. An element of a ring R is
strongly clean if it is the sum of an idempotent and a unit that commute. In
this paper, we prove that P
g∈M rgg ∈ R[M; σ] is a strongly clean element, if
re or 1 − re is strongly π-regular in R. As a corollary, we deduce that if R is a
strongly π-regular ring, then the skew monoid ring R[M; σ] is strongly clean.
These rings is a new family of non-semiprime strongly clean skew monoid rings. 

Keywords

Skew monoid ring, strongly clean ring

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