S-r-IDEALS IN COMMUTATIVE RINGS

Abstract

The rings considered in this article are commutative
with identity. This article is motivated by the
results proved by Visweswaran ([10]) on S-primary ideals.
In this paper, we introduce the concept of S-r-ideal (resp.,
S-pr-ideal) of a commutative ring and study its properties.
Let R be a commutative ring with 1 6= 0 and S be
a multiplicatively closed subset of R. Let I be an ideal
of R disjoint with S. We say that I is an S-r-ideal (resp.,
S-pr-ideal) of R if there exists an s 2 S such that for all
a, b 2 R if ab 2 I with Ann(a) = (0) implies that sb 2 I
(resp., sb 2
p
I). We investigate many properties and
characterizations of S-r-ideals (resp., S-pr-ideals). We
discuss the form of S-r-ideals (resp., S-pr-ideals) in a
finite direct product of rings. Furthermore, we study Sr-
ideals (resp., S-pr-ideals) in Nagata’s idealization ring.
Our results allow us to construct original examples of
S-r-ideals (resp., S-pr-ideals).