Special properties of differential inverse power series rings

摘要

In this paper, we continue to study the differential inverse power series ring R[[x(-1); delta]], where R is a ring equipped with a derivation delta. We characterize when R[[x(-1); delta]] is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, I-ring, respectively. Furthermore, we prove that R[[x(-1); delta]] is a domain satisfying the ACC on principal left ideals if and only if so does R. Also, for a piecewise prime ring (PWP) R we determine a large class of the differential inverse power series ring R[[x(-1); delta]] which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does R[[x(-1); delta]]. Our results extend and unify many existing results.