Baer rings and quasicontinuous rings have a MDSN

Abstract

The notion of a direct summand of a ring containing the set of nilpotents in some dense way has been considered by Y. Utumi, L. Jeremy, C. Faith, and G. F, Birkenmeier. Several types of rings including right selfinjective rings, commutative FPF rings, and rings which are a direct sum of indecomposable right ideals have been shown to have a MDSN (i.e., the minimal direct summand containing the nilpotent elements). In this paper, the class of rings which have a MDSN is enlarged to include quasiBaer rings and right quasi-continuous rings. Also, several known results are generalized. Specifically, the following results are proved: (Theorem 3) Let R be a ring in which each right annihilator of a reduced (i.e., no nonzero nilpotent elements) right ideal is essential in an idempotent generated right ideal. Then R^A®B where B is the MDSN and an essential extension of Nt (i.e., the ideal generated by the nilpotent elements of index two), and A is a reduced right ideal of R which is also an abelian Baer ring. (Corollary 6) Let R be an ATF*-aIgebra. Then R = A®B where A is a commutative AW*-algebra, and B is the MDSN of R and B is an ATF*-algebra which is a rational extension of Nt. Furthermore, A contains all reduced ideals of R. (Theorem 12) Let R be a ring such that each reduced right ideal is essential in an idempotent generated right ideal. Then R = A 0 B where B is the densely nil MDSN, and A is both a reduced quasi-continuous right ideal of R and a right quasi-continuous abelian Baer ring.