Abstract
It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.
Highlights
Introduction and backgroundEverywhere in the text of the present article, all rings are assumed to be associative, containing the identity element 1 which, in general, differs from the zero element 0
Definition 2.1. ([3]) A ring R is called strongly invo-regular if, for each r ∈ R, there exists v ∈ Inv (R) such that r = r2v, that is, r2 = rv
A ring R is said to be invo-regular if, for every r ∈ R, there exists v ∈ Inv (R) such that r = rvr, that is, r = ve, where e = vr ∈ Id (R)
Summary
Introduction and backgroundEverywhere in the text of the present article, all rings are assumed to be associative, containing the identity element 1 which, in general, differs from the zero element 0. ([3]) A ring R is called strongly invo-regular if, for each r ∈ R, there exists v ∈ Inv (R) such that r = r2v, that is, r2 = rv. It was obtained in [3, Theorem 2.2] that a ring R is strongly invo-regular if, and only if, R ⊆ λ Z2 × μ Z3 for some ordinals λ, μ.
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