CONNECTIONS BETWEEN UNIT -REGULARITY, REGULARITY, CLEANNESS, AND STRONG CLEANNESS OF ELEMENTS AND RINGS

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),, a(n) is a regular element in some ring R (for some n >= 1), then there exists w is an element of R such that a(k)w(k)a(k) = a(k) and w(k)a(k)w(k) = w(k) for 1 <= k <<= 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.