Open AccessJournal Article
A note on maximal non-prime ideals
S. Visweswaran,Anirudhdha Parmar +1 more
Reads0
Chats0
TLDR
In this article, the authors define the notion of maximal non-maximal ideal of a ring and a ring's proper ideal, i.e., a ring ideal that is maximal with respect to the property of not being a prime ideal.Abstract:
The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$, we mean an ideal $I$ of $R$ such that $Ineq R$. We say that a proper ideal $I$ of a ring $R$ is a maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$, $I$ is maximal with respect to the property of being not a prime ideal. The concept of maximal non-maximal ideal and maximal non-primary ideal of a ring can be similarly defined. The aim of this article is to characterize ideals $I$ of a ring $R$ such that $I$ is a maximal non-prime (respectively, a maximal non maximal, a maximal non-primary) ideal of $R$.read more
Citations
More filters
References
More filters
Book
Introduction to Commutative Algebra
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Journal ArticleDOI
Maximal Non-Noetherian Subrings of a Domain☆
Ahmed Ayache,Noômen Jarboui +1 more
TL;DR: The main purpose of as discussed by the authors is to study maximal non-noetherian subrings R of a domain S, and to give characterizations of such domains in several cases. But their main purpose is to give upper bounds for the number of rings and the length of chains of rings in [ R, S], the set of intermediary rings.
Journal ArticleDOI
On maximal non-accp subrings
TL;DR: In this article, a domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional non-discrete valuation domain.
Journal ArticleDOI
PID pairs of rings and maximal non-PID subrings
TL;DR: The first purpose of this paper is to study maximal non-PID subrings and characterize these type of rings.