ON 2-ABSORBING PRIMARY IDEALS IN COMMUTATIVE RINGS

Badawi, Ayman,Tekir, Unsal,Yetkin, Ece Korean Mathematical Society 2014 대한수학회보 Vol.51 No.4

Let R be a commutative ring with $1{\neq}0$. 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 $a,b,c{\in}R$ and $abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

Weakly Classical Prime Submodules

Mostafanasab, Hojjat,Tekir, Unsal,Oral, Kursat Hakan Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.4

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each $m{\in}M$ and elements a, $b{\in}R$, $abm{\in}N$ implies that $am{\in}N$ or $bm{\in}N$. We introduce the concept of "weakly classical prime submodules" and we will show that this class of submodules enjoys many properties of weakly 2-absorbing ideals of commutative rings. A proper submodule N of M is a weakly classical prime submodule if whenever $a,b{\in}R$ and $m{\in}M$ with $0{\neq}abm{\in}N$, then $am{\in}N$ or $bm{\in}N$.

On 2-absorbing primary ideals in commutative rings

Ayman Badawi,Unsal Tekir,Ece Yetkin 대한수학회 2014 대한수학회보 Vol.51 No.4

Let R be a commutative ring with 1 6= 0. In this paper, we introduce the concept of 2-absorbing primary ideal which is a general- ization 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.

ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

Ayman Badawi,Unsal Tekir,Ece Yetkin 대한수학회 2015 대한수학회지 Vol.52 No.1

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

On strongly quasi primary ideals

Suat Koc,Unsal Tekir,Gulsen Ulucak 대한수학회 2019 대한수학회보 Vol.56 No.3

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let $R$ be a commutative ring with nonzero identity and $Q$ a proper ideal of $R$. Then $Q$ is called strongly quasi primary if $ab\in Q$ for $a,b\in R$ implies either $a^{2}\in Q$ or $b^{n}\in Q~ (a^{n}\in Q$ or $b^{2}\in Q)$ for some $n\in \mathbb{N} $. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph $\Gamma_{I}(R)$ and denote it by $\Gamma_{I}^{\ast}(R)$, where $I$ is an ideal of $R$. We investigate the relations between $\Gamma_{I}^{\ast} (R)$ and $\Gamma_{I}(R)$. Further, we use strongly quasi primary ideals and $\Gamma_{I}^{\ast}(R)$ to characterize von Neumann regular rings.

ON STRONGLY QUASI PRIMARY IDEALS

Koc, Suat,Tekir, Unsal,Ulucak, Gulsen Korean Mathematical Society 2019 대한수학회보 Vol.56 No.3

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.

ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

Badawi, Ayman,Tekir, Unsal,Yetkin, Ece Korean Mathematical Society 2015 대한수학회지 Vol.52 No.1

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

COMMUTATIVE RINGS AND MODULES THAT ARE r-NOETHERIAN

Anebri, Adam,Mahdou, Najib,Tekir, Unsal Korean Mathematical Society 2021 대한수학회보 Vol.58 No.5

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring and M be an R-module. We say that M is an r-Noetherian module if every r-submodule of M is finitely generated. Also, we call the ring R to be an r-Noetherian ring if R is an r-Noetherian R-module, or equivalently, every r-ideal of R is finitely generated. We show that many properties of Noetherian modules are also true for r-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of r-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of r-Noetherian rings that are not Noetherian.

$S$-versions and $S$-generalizations of idempotents, pure ideals and Stone type theorems

Bayram Ali Ersoy,Unsal Tekir,Eda Yildiz 대한수학회 2024 대한수학회보 Vol.61 No.1

Let $R$ be a commutative ring with nonzero identity and $M$ be an $R$-module. In this paper, we first introduce the concept of $S$-idempotent element of $R$. Then we give a relation between $S$-idempotents of $R$ and clopen sets of $S$-Zariski topology. After that we define $S$-pure ideal which is a generalization of the notion of pure ideal. In fact, every pure ideal is $S$-pure but the converse may not be true. Afterwards, we show that there is a relation between $S$-pure ideals of $R$ and closed sets of $S$-Zariski topology that are stable under generalization.

ON WEAKLY 2-ABSORBING PRIMARY SUBMODULES OF MODULES OVER COMMUTATIVE RINGS

Ahmad Yousefian Darani,Fatemeh Soheilnia,Unsal Tekir,Gulsen Ulucak 대한수학회 2017 대한수학회지 Vol.54 No.5

Assume that $M$ is an $R$-module where $R$ is a commutative ring. A proper submodule $N$ of $M$ is called a weakly $2$-absorbing primary submodule of $M $ if $0\neq abm\in N$ for any $a,b\in R$ and $m\in M$, then $ab\in (N:M)$ or $am\in M\mbox{-rad}(N)$ or $bm\in M\mbox{-rad}(N).$ In this paper, we extended the concept of weakly $2$-absorbing primary ideals of commutative rings to weakly $2$-absorbing primary submodules of modules. Among many results, we show that if $N$ is a weakly $2$-absorbing primary submodule of $ M$ and it satisfies certain condition $0\neq I_{1}I_{2}K\subseteq N$ for some ideals $I_{1},I_{2}$ of $R$ and submodule $K$ of $M$, then $ I_{1}I_{2}\subseteq (N:M)$ or $I_{1}K\subseteq M\mbox{-rad}(N)$ or $ I_{2}K\subseteq M\mbox{-rad}(N)$.