Strongly primary ideals in rings with zero-divisors

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.

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.

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.

Journal of Algebra and Its Applications

Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce the concept of 1-absorbing primary ideals in commutative rings. A proper ideal [Formula: see text] of [Formula: see text] is called a [Formula: see text]-absorbing primary ideal of [Formula: see text] if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some properties of 1-absorbing primary ideals are investigated. For example, we show that if [Formula: see text] admits a 1-absorbing primary ideal that is not a primary ideal, then [Formula: see text] is a quasilocal ring. We give an example of a 1-absorbing primary ideal of [Formula: see text] that is not a primary ideal of [Formula: see text]. We show that if [Formula: see text] is a Noetherian domain, then [Formula: see text] is a Dedekind domain if and only if every nonzero proper 1-absorbing primary ideal of [Formula: see text] is of the form [Formula: see text] fo...

Discussiones Mathematicae - General Algebra and Applications

In this paper, we define quasi-primary ideals in commutative semirings S with 1 = 0 which is a generalization of primary ideals. A proper ideal I of a semiring S is said to be a quasi-primary ideal of We also introduce the concept of 2-absoring quasi-primary ideal of a semiring S which is a generalization of quasi-primary ideal of S. A proper ideal I of a semiring S is said to be a 2-absorbing quasi-primary ideal if abc ∈ √ I implies ab ∈ √ I or bc ∈ √ I or ac ∈ √ I. Some basic results related to 2-absorbing quasi-primary ideal have also been given.

Journal of Pure and Applied Algebra, 2008

This article introduces and advances the basic theory of "uniformly primary ideals" for commutative rings, a concept that imposes a certain boundedness condition on the usual notion of "primary ideal". Characterizations of uniformly primary ideals are provided along with examples that give the theory independent value. Applications are also provided in contexts that are relevant to Noetherian rings.

We present *-primary submodules, a generalization of the concept of primary submodules of an R-module. We show that every primary submodule of a Noetherian R-module is *-primary. Among other things, we show that over a commutative domain R, every torsion free R-module is *-primary. Furthermore, we show that in a cyclic R-module, primary and *-primary coincide. Moreover, we give a characterization of *-primary submodules for some finitely generated free R-modules.

2020

Let R be a commutative ring with 1 6= 0. In this paper, we introduce a subclass of the class of 1-absorbing primary ideals called the class of strongly 1-absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ √ 0. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the UN -rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.

Bulletin of the Korean Mathematical Society, 2014

Let R be a commutative ring with 1 = 0. In this paper, we introduce the concept of 2-absorbing primary ideal which is a generalization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ √ I or bc ∈ √ I. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

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.