Thus, for x EI we get and the xi's generate I. We show this to be the case for J. Notice ICJ; if J
is not A, let M be a maximal ideal containing it. By hypothesis M is not a minimal prime ideal …

Let R be an integral domain in which all finitely generated ideals are principal. Let M be a
projective R-module. It is well known that if M is finitely generated it is free. By a familiar …

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and? is a
prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also …

Let $ R $ be a commutative ring with identity, let $ X $ be an indeterminate, and let $ I $ be
an ideal of the polynomial ring $ R [X] $. Let $\min I $ denote the set of elements of $ I $ of …

PM Cohn [7] calls a submodule P of the left A-module M pure iff 0→ E⊗ P→ E⊗ M is exact for
all right modules E. This definition of purity, which Cohn [7] has shown to be equivalent to …

Introduction. The aim of this paper is to study conditions which reflect the projectivity of a
given finitely generated flat module over a commutative ring. The use of the invariant factors …

Let $ R $ be a commutative ring with identity $1\ne 0$ and let $ A $ be a nonzero ideal of $
R $. A problem of current interest is to relate the notions of “projective ideal",“flat ideal” and …

A module is serial if its submodules are linearly ordered with respect to inclusion. A ring R is
called left serial if RR is a direct sum of serial modules, and R is called serial if it is both left …

The main results in this paper relate the concepts of flatness and projectiveness for finitely
generated ideals in a commutative ring with unity. In this discussion the idea of a …

In this paper, I will be a regular ideal in a Noetherian ring R and Ī will denote the integral
closure of I. If J is another ideal of I. and J are said to be projectively equivalent if there are …