scholarly journals 1-absorbing primary submodules

2021 ◽  
Vol 29 (3) ◽  
pp. 285-296
Author(s):  
Ece Yetkin Celikel
Keyword(s):  
Commutative Ring ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Primary Ideals

Abstract Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N : RM) or m ∈ M − rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

scholarly journals On 2-Absorbing Primary Submodules of Modules over Commutative Rings

2016 ◽  
Vol 24 (1) ◽  
pp. 335-351 ◽  
Author(s):  
Hojjat Mostafanasab ◽  
Ece Yetkin ◽  
Ünsal Tekir ◽  
Ahmad Yousefian Darani
Keyword(s):  
Commutative Rings ◽  
Primary Ideal ◽  
Finitely Generated ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Prime Submodule ◽  
Primary Ideals

Abstract All rings are commutative with 1 ≠ 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a; b ∈ R and m ∈ M and abm ∈ N, then am ∈ M-rad(N) or bm ∈ M-rad(N) or ab ∈(N :R M). It is shown that a proper submodule N of M is a 2-absorbing primary submodule if and only if whenever I1I2K ⊆ N for some ideals I1; I2 of R and some submodule K of M, then I1I2 ⊆ (N :R M) or I1K ⊆ M-rad(N) or I2K ⊆ M-rad(N). We prove that for a submodule N of an R-module M if M-rad(N) is a prime submodule of M, then N is a 2-absorbing primary submodule of M. If N is a 2-absorbing primary submodule of a finitely generated multiplication R-module M, then (N :R M) is a 2-absorbing primary ideal of R and M-rad(N) is a 2-absorbing submodule of M.

scholarly journals On 2-absorbing quasi primary submodules

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2943-2950 ◽  
Author(s):  
Suat Koc ◽  
Uregen Nagehan ◽  
Unsal Tekir
Keyword(s):  
Commutative Ring ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity, and let M be a nonzero unital R-module. In this article, we introduce the concept of 2-absorbing quasi primary submodules which is a generalization of prime submodules. We define 2-absorbing quasi primary submodule as a proper submodule N of M having the property that abm ? N, then ab ? ?(N :R M) or am ? radM(N) or bm ? radM(N): Various results and examples concerning 2-absorbing quasi primary submodules are given.

scholarly journals Approximaitly Primary Submodules

2019 ◽  
Vol 24 (5) ◽  
pp. 105
Author(s):  
Ali Sh. Ajeel ◽  
Haibat K. Mohammadali
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
The Other ◽  
Left Module ◽  
Basic Properties ◽  
Other Hand ◽  
Prime Submodules ◽  
Primary Submodules ◽  
Primary Submodule

The study deals with the notion of an approximaitly primary submodules of unitary left -module  over a commutative ring  with identity as a generalization of a primary submodules and approximaitly prime submodules, where a proper submodule  of an -module  is called an approximaitly primary submodule of , if whenever , for , , implies that either  or  for some positive integer  of . Several characterizations, basic properties of this concept are given. On the other hand the relationships of this concept with some classes of modules are studied. Furthermore, the behavior of approximaitly primary submodule under -homomorphism are discussed   http://dx.doi.org/10.25130/tjps.24.2019.098

On 2-absorbing primary submodules of modules over commutative ring with unity

2015 ◽  
Vol 08 (04) ◽  
pp. 1550064 ◽  
Author(s):  
Manish Kant Dubey ◽  
Pakhi Aggarwal
Keyword(s):  
Commutative Ring ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Nonzero Identity

In this paper, we introduce the concept of a [Formula: see text]-absorbing primary submodule over a commutative ring with nonzero identity which is a generalization of primary submodule. Let [Formula: see text] be an [Formula: see text]-module and [Formula: see text] be a proper submodule of [Formula: see text]. Then [Formula: see text] is said to be a [Formula: see text]-absorbing primary submodule of [Formula: see text] if whenever [Formula: see text], [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] or [Formula: see text]. We have given an example and proved number of results concerning [Formula: see text]-absorbing primary submodules. We have also proved the [Formula: see text]-absorbing primary avoidance theorem for submodules.

Primary ideals with finitely generated radical in a commutative ring

1993 ◽  
Vol 78 (1) ◽  
pp. 201-221 ◽  
Author(s):  
Robert Gilmer ◽  
William Heinzer
Keyword(s):  
Commutative Ring ◽  
Finitely Generated ◽  
Primary Ideals

scholarly journals On Generalizations of phi-2-Absorbing Primary Submodules

2018 ◽  
Vol 11 (1) ◽  
pp. 35
Author(s):  
Pairote Yiarayong ◽  
Manoj Siripitukdet
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Primary Submodules ◽  
Primary Submodule ◽  
Necessary And Sufficient

Let $\phi: S(M) \rightarrow S(M) \cup \left\lbrace \emptyset\right\rbrace $ be a function where $S(M)$ is the set of all submodules of $M$. In this paper, we extend the concept of $\phi$-$2$-absorbing primary submodules to the context of $\phi$-$2$-absorbing semi-primary submodules. A proper submodule $N$ of $M$ is called a $\phi$-$2$-absorbing semi-primary submodule, if for each $m \in M$ and $a_{1}, a_{2}\in R$ with $a_{1}a_{2}m \in N - \phi(N)$, then $a_{1}a_{2}\in \sqrt{(N : M)}$ or  $a_{1}m \in N$ or $a^{n}_{2}m\in N$, for some positive integer $n$. Those are extended from $2$-absorbing primary, weakly $2$-absorbing primary, almost $2$-absorbing primary, $\phi_{n}$-$2$-absorbing primary, $\omega$-$2$-absorbing primary and $\phi$-$2$-absorbing primary submodules, respectively. Some characterizations of $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-$2$-absorbing semi-primary submodules are obtained. Moreover, we investigate relationships between $2$-absorbing semi-primary, $\phi_{n}$-$2$-absorbing semi-primary and $\phi$-primary submodules of modules over commutative rings. Finally, we obtain necessary and sufficient conditions of a $\phi$-$\phi$-$2$-absorbing semi-primary in order to be a $\phi$-$2$-absorbing semi-primary.

scholarly journals On graded primary-like submodules of graded modules over graded commutative rings

2021 ◽  
Author(s):  
Khaldoun Falah Al-Zoubi ◽  
Mohammed Al-Dolat
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Graded Modules ◽  
Primary Ideals

Let G be a group with identity e. Let R be a G-graded commutative ring andM a graded R-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to and extra properties of these graded submodules.

scholarly journals On 1-absorbing primary ideals of commutative rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050111 ◽  
Author(s):  
Ayman Badawi ◽  
Ece Yetkin Celikel
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Primary Ideal ◽  
Ideal Formula ◽  
Noetherian Domain ◽  
Primary Ideals ◽  
Nonzero Identity

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] for some nonzero prime ideal [Formula: see text] of [Formula: see text] and a positive integer [Formula: see text]. We show that a proper ideal [Formula: see text] of [Formula: see text] is a 1-absorbing primary ideal of [Formula: see text] if and only if whenever [Formula: see text] for some proper ideals [Formula: see text] of [Formula: see text], then [Formula: see text] or [Formula: see text]

scholarly journals Reverse mathematics of rings

2021 ◽  
Author(s):  
◽  
Jordan Barrett
Keyword(s):  
Commutative Ring ◽  
Systematic Study ◽  
Reverse Mathematics ◽  
Ring Theory ◽  
Base System ◽  
Weak Base ◽  
Integral Domains ◽  
Fine Grained ◽  
Order Arithmetic ◽  
Primary Ideals

<p>Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system RCA_0 + Sigma-2 induction. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bézout and GCD domains.</p>

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