It is well known that the set of nilpotent elements of a commutative ring is the intersection of
all the prime ideals. A generalization of this to modules over a commutative ring was …

We will state some conditions under which if a quotient of a module M satisfies the radical
formula of degree k (strf of degree k), so does M. Especially, we will introduce some …

Throughout this note, all rings are commutative and all modules are unitary. Let R be a ring
and M an R-module. We shall define the prime radical of M to be the intersection of M and all …

It is well known that in a commutative Noetherian ring, a prime ideal minimal over the
annihilator of a finitely generated module is the annihilator of some element of that module …

In a commutative ring R the radical of an ideal A is defined as the set of all x in R such that
x~ eA for some natura] number n and this definition makes the radical an ideal. This is …

Throughout, R will denote a commutative ring with identity and every module is unitary. By
B< A we mean that B is a proper submodule of A. For the submodules B and C of A, we let …

In this paper we describe the radical of a finitely generated submodule N of a free module F
of finite rank. As a result of this characterization we can obtain the radical of a finitely …

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 …

The radical of a module over a commutative ring is the intersection of all prime submodules.
It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is …

The coincidence of the set of all nilpotent elements of a ring with its prime radical has a
module analogue which occurs when the zero submodule satisfies the radical formula. A …