Generalizations of Prime Ideals

2014

Abstract. Let R be a commutative ring with identity. Let ϕ: I(R) → I(R) ∪ {∅} be a function where I(R) denotes the set of all ideals of R. A proper ideal Q of R is called ϕ-primary if whenever a, b ∈ R, ab ∈ Q−ϕ(Q) implies that either a ∈ Q or b ∈ √ Q. So if we take ϕ∅(Q) = ∅ (resp., ϕ0(Q) = 0), a ϕ-primary ideal is primary (resp., weakly primary). In this paper we study the properties of several generalizations of primary ideals of R. AMS Mathematics Subject Classification (2010): 13A15 Key words and phrases: primary ideal, weakly primary ideal, almost primary ideal, ϕ-primary ideal, strongly primary ideal 1.

Communications in Algebra, 2000

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.

Quaestiones Mathematicae, 2020

Let A be an integral domain with quotient field K. A. Badawi and E. Houston called a strongly primary ideal I of A if whenever x, y ∈ K and xy ∈ I, we have x ∈ I or y n ∈ I for some n ≥ 1. In this note, we study the generalization of strongly primary ideal to the context of arbitrary commutative rings. We define a primary ideal P of A to be strongly primary if for each a, b ∈ A, we have aP ⊆ bA or b n A ⊆ a n P for some n ≥ 1.

Formalized Mathematics, 2021

Summary. We formalize in the Mizar System [3], [4], definitions and basic propositions about primary ideals of a commutative ring along with Chapter 4 of [1] and Chapter III of [8]. Additionally other necessary basic ideal operations such as compatibilities taking radical and intersection of finite number of ideals are formalized as well in order to prove theorems relating primary ideals. These basic operations are mainly quoted from Chapter 1 of [1] and compiled as preliminaries in the first half of the article.

Communications in Algebra

In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring are obtained. Some applications of this study are also given. In particular, the new notion of semi-noetherian ring is introduced and Cohen type theorem is proved.

2008

The primary purpose of this paper is give a classification scheme for the nonzero primes of a Prüfer domain based on five properties. A prime P of a Prüfer domain R could be sharp or not sharp, antesharp or not, divisorial or not, branched or unbranched, idempotent or not. Based on these five basic properties, there are six types of maximal ideals and twelve types of nonmaximal (nonzero) primes. Both characterizations and examples are given for each type that exists.

2011

In this paper a new type of ideals in commutative rings is defined which iscalled an almost primary ideal. Some properties of this type of ideals are obtained and also, some characterizations of them are given.

Transactions of the American Mathematical Society, 1965

A commutative ring R is called an AM-ring (for allgemeine multiplikationsring) if whenever A and B are ideals of R with A properly contained in B, then there is an ideal C of R such that A = BC. An AM-ring R in which

Bulletin of The Australian Mathematical Society, 1993

Let R be a commutative ring with identity and all modules are unital. Various generalizations of primary ideals and primary modules have been studied. For example, a proper ideal I of R is weakly primary if whenever 0=ab∈I, then a ∈I or b∈ Rad (I). Also a proper submodule N of R-module M is weakly primary submodule if whenever r∈R and m∈M such that 0≠rm∈N , then either m ∈N or r n ∈(N:M). Through out this work, we define almost primary ideals and almost primary submodules as a new generalization of weakly primary ideals (resp., primary submodules). We show that weakly primary ideals (resp., primary submodules) enjoy analogs of many of the properties of primary ideals (resp., submodules).