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 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.

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.

$S$-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.

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.

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 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.

Comparison of the median and intermediate approaches to the ultrasound-guided sacral erector spinae plane block: a cadaveric and radiologic study

Keleş Bilge Olgun,Salman Necati,Yılmaz Elvan Tekir,Birinci Habip Resul,Apan Alparslan,İnce Selami,Özyaşar Ali Faruk,Uz Aysun 대한마취통증의학회 2024 Korean Journal of Anesthesiology Vol.77 No.1

Background: Erector spinae plane block (ESPB) is a well-established method for managing postoperative and chronic pain. ESPB applications for the sacral area procedures are called sacral ESPBs (SESPBs). This cadaveric study aimed to determine the distribution of local anesthesia using the median and intermediate approaches to the SESPB.Methods: Four cadavers were categorized into the median and intermediate approach groups. Ultrasound-guided SESPBs were performed using a mixture of radiopaque agents and dye. Following confirmation of the solution distribution through computed tomography (CT), the cadavers were dissected to observe the solution distribution.Results: CT images of the median group demonstrated subcutaneous pooling of the radiopaque solution between the S1 and S5 horizontal planes. Radiopaque solution also passed from the sacral foramina to the anterior sacrum via the spinal nerves between S2 and S5. In the intermediate group, the solution distribution was observed along the bilateral erector spinae muscle between the L2 and S3 horizontal planes; no anterior transition was detected. Dissection in the median group revealed blue solution distribution in subcutaneous tissue between horizontal planes S1 and S5, but no distribution in superficial fascia or muscle. In the intermediate group, red solution was detected in the erector spinae muscle between the L2 and S3 intervertebral levels.Conclusions: Radiologic and anatomic findings revealed the presence of radiopaque dye in the superficial and erector spinae compartments in both the median and intermediate groups. However, anterior transition of the radiopaque dye was detected only in the median group.