Connections between unit-regularity, regularity, cleanness, and strong cleanness of elements and rings

Abstract:

We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. Next we study in arbitrary rings those elements whose powers are regular, and provide a method for constructing inner inverses which satisfy many additional strong relations. As a corollary we show that if each of the powers $a,a^2,\ldots , a^n$ is a regular element in some ring $R$ (for some $n\geq 1$), then there exists $w\in R$ such that $a^k w^k a^k=a^k$ and $w^k a^k w^k=w^k$ for $1\leq k\leq n$. Similar statements are also obtained for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanness, regularity, and unit-regularity.