Group-regular rings

We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one. We also prove that these rings are morphic rings.

Walker's Cancellation Theorem

Walker's cancellation theorem says that if B + Z isisomorphic to C + Z in the category of abeliangroups, then B is isomorphic to C. We construct an example ina diagram category of abelian groups where the theorem fails. As aconsequence, the original theorem does not have a constructiveproof. In fact, in our example B and C are subgroups ofZ<sup>2</sup>. Both of these results contrast with a group whoseendomorphism ring has stable range one, which allows aconstructive proof of cancellation and also a proof in any diagramcategory.

Special properties of differential inverse power series rings

In this paper, we continue to study the differential inverse power series ring [Formula: see text], where [Formula: see text] is a ring equipped with a derivation [Formula: see text]. We characterize when [Formula: see text] is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, [Formula: see text]-ring, respectively. Furthermore, we prove that [Formula: see text] is a domain satisfying the [Formula: see text] on principal left ideals if and only if so does [Formula: see text]. Also, for a piecewise prime ring (PWP) [Formula: see text] we determine a large class of the differential inverse power series ring [Formula: see text] which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if [Formula: see text] has a (flat) projective socle, then so does [Formula: see text]. Our results extend and unify many existing results.

SUMS OF UNITS IN RINGS

In this paper, we study rings that are additively generated by units. We prove that if the identity in a ring with stable range one is a sum of two units, then every (von Neumann) regular element is a sum of two units. It follows that every element in a unit-regular ring is a sum of two units if the identity is a sum of two units. Also, if the identity of a strongly π-regular ring is a sum of two units, then every element is a sum of three units.