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 self-injective 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 quasi-Baer 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 AW∗-aIgebra. Then R = A ⊕ B
where A is a commutative AW∗-algebra, and B is the MDSN of R and
B is an AW∗-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 ⊕ 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.
