The term The Theory of Rings seems first used as title of a book by Jacobson [43], and in his
preface Jacobson asserts that the theory that forms the subject of the book had its beginning …

Page 1 PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks LINEAR
ALGEBRA OVER COMMUTATIVE RINGS Bernard R. McDonald Page 2 Linear Algebra Over …

A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the
last 30 years several characterizations of commutative clean rings have been given. We …

The purpose of this paper is to investigate pure submodules of multiplication modules. We
introduce the concept of idempotent submodule generalizing idempotent ideal. We show …

IA very interesting theorem of Bkouche ([Bk], Th. 6) characterizes projectivity of pure ideals in
terms of the topology of Max (A), when A is a «soft» ring. The original purpose of this paper …

<[N= I< theta( M) M=< theta( M)( IM)=< theta( M) N< noindent If $ N $ is finitely generated
then$ R=< theta( M)+ $ ann$ N,[6, $ Theorem 76].< quad If moreover $ N $ is faithful then< …

Let CF (X) denote the socle of C (X). It is shown that X is a P-space if and only if C (X) is a ℵ0-
selfinjective ring or equivalently, if and only if is ℵ0-selfinjective. We also prove that X is an …

In this paper, new algebraic and topological results on purely-prime ideals of a commutative
ring (pure spectrum) are obtained. Particularly, Grothendieck-type theorem is obtained …

Let C be a class of modules and L= lim→ C the class of all direct limits of modules from C.
The class L is well understood when C consists of finitely presented modules: L then enjoys …

All rings considered in this paper are commutative with unity. A ring R is called a PF-ring if
every principal ideal a R is a flat R-module, and it is called a PP-ring if every principal ideal a …