# Classical 1-Absorbing Primary Submodules

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## Abstract

**:**

## 1. Introduction

## 2. Characterizations of Classical 1-Absorbing Primary Submodules

**Definition 1.**

**Example 1.**

**Theorem 1.**

- 1.
- K is a classical 1-a.p submodule of M;
- 2.
- For every $\eta \in M\setminus K$, $\left(K{:}_{\Re}\eta \right)$ is a 1-a.p ideal of ℜ;
- 3.
- For every $\eta \in M\setminus K$, $\left(K{:}_{\Re}\eta \right)$ is a 1-a.p ideal of ℜ and $\left\{\sqrt{\left(K{:}_{\Re}\eta \right)}:\eta \in M\setminus K\right\}$ is a chain of prime ideals of ℜ.

**Proof.**

**Lemma 1.**

**Proof.**

**Example 2.**

**Theorem 2.**

- 1.
- K is a 1-a.p submodule;
- 2.
- K is a classical 1-a.p submodule;
- 3.
- $\left(K{:}_{\Re}M\right)$ is a 1-a.p ideal of ℜ.

**Proof.**

**Theorem 3.**

**Proof.**

**Theorem 4.**

- 1.
- K is a classical 1-a.p submodule of $M;$
- 2.
- For each nonunits $x,y,z\in \Re $, $\left(K{:}_{M}xyz\right)\subseteq $ $\left(K{:}_{M}xy\right)\cup \left({\cup}_{t\ge 1}\left(K{:}_{M}{z}^{t}\right)\right)$;
- 3.
- For each nonunits $x,y\in \Re $ and $\eta \in M$ with $xy\eta \notin K$, $\left(K{:}_{\Re}xy\eta \right)\subseteq \sqrt{\left(K{:}_{\Re}\eta \right)}$;
- 4.
- For each nonunits $x,y\in \Re $ and every ideal J of ℜ and $\eta \in M$ with $xyJ\eta \subseteq K$, either $xy\eta \in K$ or $J\subseteq \sqrt{\left(K{:}_{\Re}\eta \right)}$;
- 5.
- For each ideals $I,J,L$ of ℜ and $\eta \in M$ with $IJL\eta \subseteq K$, either $IJ\eta \subseteq K$ or $L\subseteq \sqrt{\left(K{:}_{\Re}\eta \right)}.$

**Proof.**

**Theorem 5.**

- 1.
- K is a classical 1-a.p submodule of M;
- 2.
- If $IJL\eta \subseteq K$ for some proper submodules $I,J,L$ of M and $\eta \in M$, then either $IJ\eta \subseteq K$ or ${L}^{t}\eta \subseteq K$ for some $t\ge 1$.

**Proof.**

**Remark 1.**

**Proof.**

**Proposition 1.**

- 1.
- For all nonunits $x,y,z\in \Re $ and $\eta \in M$, $\left(K{:}_{\Re}xyz\eta \right)\subseteq $$\left(K{:}_{\Re}xy\eta \right)\cup \left({\cup}_{t\ge 1}\left(K{:}_{\Re}{z}^{t}\eta \right)\right).$
- 2.
- If ℜ is a u-ring, then, for all nonunits $x,y,z\in \Re $ and $\eta \in M$, $\left(K{:}_{\Re}xyz\eta \right)\subseteq $$\left(K{:}_{\Re}xy\eta \right)$ or $\left(K{:}_{\Re}xyz\eta \right)\subseteq $$\left(K{:}_{\Re}{z}^{t}\eta \right)$ for some $t\ge 1$.

**Proof.**

**Theorem 6.**

- 1.
- K is a classical 1-a.p submodule of $M;$
- 2.
- For all nonunits $x,y,z\in \Re $, $\left(K{:}_{M}xyz\right)=\left(K{:}_{M}xy\right)$ or $\left(K{:}_{M}xyz\right)\subseteq \left(N{:}_{M}{z}^{t}\right)$ for some $t\ge 1$;
- 3.
- For all nonunits $x,y,z\in \Re $ and every submodule L of M; $xyzL\subseteq K$ implies that $xyL\subseteq K$ or ${z}^{t}L\subseteq K$ for some $t\ge 1$;
- 4.
- For all nonunits $x,y\in \Re $ and every submodule L of M with $xyL\u2288K$, $\left(K{:}_{\Re}xyL\right)\subseteq \sqrt{\left(K{:}_{\Re}L\right)}$;
- 5.
- For all nonunits $x,y\in \Re $, every ideal J of ℜ and every submodule L of M with $xyJL\subseteq K$, this implies that $xyL\subseteq K$ or $J\subseteq \sqrt{\left(K{:}_{\Re}L\right)}$;
- 6.
- For all ideals $I,J,P$ of ℜ and every submodule L of M with $IJPL\subseteq K$, this implies that $IJL\subseteq K$ or $P\subseteq \sqrt{\left(K{:}_{\Re}L\right)}.$

**Proof.**

**Theorem 7.**

- 1.
- K is a classical 1-a.p submodule of M;
- 2.
- If ${K}_{1}{K}_{2}{K}_{3}{K}_{4}\subseteq K$ for some submodules ${K}_{1},{K}_{2},{K}_{3}$$,{K}_{4}$ of M, then either ${K}_{1}{K}_{2}{K}_{4}\subseteq K$ or ${K}_{3}^{t}{K}_{4}\subseteq K$ for some $t\ge 1$;
- 3.
- If ${K}_{1}{K}_{2}{K}_{3}\subseteq K$ for some submodules ${K}_{1},{K}_{2},{K}_{3}$ of M, then either ${K}_{1}{K}_{2}\subseteq K$ or ${K}_{3}^{t}\subseteq K$ for some $t\ge 1$;
- 4.
- K is a 1-a.p submodule of M;
- 5.
- $\left(K{:}_{R}M\right)$ is a 1-a.p ideal of ℜ.

**Proof.**

**Proposition 2.**

- 1.
- If K is a classical 1-a.p submodule of ℜ-module M, such that $\left(K{:}_{\Re}M\right)\cap S=\varnothing $, then ${S}^{-1}K$ is a classical 1-a.p submodule of ${S}^{-1}M$.
- 2.
- If ${S}^{-1}K$ is a classical 1-a.p submodule of ${S}^{-1}M$ and ${Z}_{\Re}(M/K)\cap S=\varnothing $, then K is a classical 1-a.p submodule of $M.$

**Proof.**

**Proposition 3.**

- 1.
- If ${K}_{2}$ is a classical 1-a.p submodule of ${M}_{2}$ and ${f}^{-1}\left({K}_{2}\right)$ is proper, then ${f}^{-1}\left({K}_{2}\right)$ is a classical 1-a.p submodule of ${M}_{1}$.
- 2.
- If f is an epimorphism and ${K}_{1}$ is a classical 1-a.p submodule of ${M}_{1}$ containing $Ker\left(f\right)$, then $f\left({K}_{1}\right)$ is a classical 1-a.p submodule of ${M}_{2}$.

**Proof.**

**Corollary 1.**

**Proposition 4.**

**Proof.**

**Corollary 2.**

**Proof.**

**Proposition 5.**

**Proof.**

**Example 3.**

**Proposition 6.**

- 1.
- ${K}_{1}$ is a 1-absorbing prime submodule of ${M}_{1}$ and ${K}_{2}={M}_{2}$;
- 2.
- ${K}_{2}$ is a 1-absorbing prime submodule of ${M}_{2}$ and ${K}_{1}={M}_{1}.$

**Proof.**

**Example 4.**

**Theorem 8.**

**Proof.**

## 3. Classical Primary-like Conditions in Amalgamated Duplication of a Module

**Lemma 2.**

- 1.
- $(K\bowtie J{:}_{M\bowtie J}(x,x+j))=\left(K{:}_{M}x\right)\bowtie J$ for every $x\in \Re $ and $j\in J;$
- 2.
- $(K\bowtie J{:}_{\Re \bowtie J}(\eta ,\eta +{\eta}^{\prime}))=\left(K{:}_{\Re}\eta \right)\bowtie J$ for every $\eta \in M$ and ${\eta}^{\prime}\in JM.$

**Proof.**

**Lemma 3**

**.**Consider $M$ as an ℜ-module and $K\phantom{\rule{4pt}{0ex}}$ a submodule of $M.\phantom{\rule{4pt}{0ex}}$Then, K is a classical prime submodule if and only if $\left(K{:}_{M}xy\right)=(K:x)$ or $\left(K{:}_{M}xy\right)=\left(K{:}_{M}y\right)$ for every $x,y\in \Re .\phantom{\rule{4pt}{0ex}}$

**Lemma 4.**

**Proof.**

**Theorem 9.**

- 1.
- $K\bowtie J$ is a classical primary submodule of $M\bowtie J$ if and only if K is a classical primary submodule of $M;$
- 2.
- $K\bowtie J$ is a classical 1-a.p submodule of $M\bowtie J$ if and only if K is a classical 1-a.p submodule of $M.\phantom{\rule{4pt}{0ex}}$

**Proof.**

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Yılmaz Uçar, Z.; Ersoy, B.A.; Tekir, Ü.; Yetkin Çelikel, E.; Onar, S.
Classical 1-Absorbing Primary Submodules. *Mathematics* **2024**, *12*, 1801.
https://doi.org/10.3390/math12121801

**AMA Style**

Yılmaz Uçar Z, Ersoy BA, Tekir Ü, Yetkin Çelikel E, Onar S.
Classical 1-Absorbing Primary Submodules. *Mathematics*. 2024; 12(12):1801.
https://doi.org/10.3390/math12121801

**Chicago/Turabian Style**

Yılmaz Uçar, Zeynep, Bayram Ali Ersoy, Ünsal Tekir, Ece Yetkin Çelikel, and Serkan Onar.
2024. "Classical 1-Absorbing Primary Submodules" *Mathematics* 12, no. 12: 1801.
https://doi.org/10.3390/math12121801