# A Note on Weakly S-Noetherian Rings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic properties

**Proposition**

**1.**

- (1)
- If R is a weakly S-Noetherian ring which contains a nonunit regular element, then R is an S-Noetherian ring. In particular, if R is an integral domain, then R is a weakly S-Noetherian ring if and only if R is an S-Noetherian ring.
- (2)
- R is a weakly S-Noetherian ring in which every maximal ideal is S-finite if and only if R is an S-Noetherian ring.

**Proof**.

**Corollary**

**1.**

- (1)
- R is an S-Noetherian ring and T is an S-finite R-module if and only if $R+XT\left[X\right]$ is a weakly S-Noetherian ring.
- (2)
- R is an S-Noetherian ring if and only if $R\left[X\right]$ is a weakly S-Noetherian ring.

**Proof**.

**Corollary**

**2.**

- (1)
- R is an S-Noetherian ring and T is an S-finite R-module if and only if $R+XT\left[\phantom{\rule{-0.166667em}{0ex}}\right[X\left]\phantom{\rule{-0.166667em}{0ex}}\right]$ is a weakly S-Noetherian ring.
- (2)
- R is an S-Noetherian ring if and only if $R\left[\phantom{\rule{-0.166667em}{0ex}}\right[X\left]\phantom{\rule{-0.166667em}{0ex}}\right]$ is a weakly S-Noetherian ring.

**Proof**.

**Example**

**1.**

- (1)
- For each $i\ge 2$ and $j\ge 1$, let$${a}_{ij}=\left\{\begin{array}{cc}p\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}i=j\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$
- (2)
- Let I be an ideal of R. Then $(1,p,p,0,0,\cdots )I$ is a finite ideal of R; so I is a T-finite ideal of R. Hence R is a T-Noetherian ring. Thus R is a weakly T-Noetherian ring.

**Example**

**2.**

- (1)
- Let ${A}_{1}$ be the ideal of R generated by the set $\left\{\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{i}}\phantom{\rule{0.166667em}{0ex}}\right|\phantom{\rule{0.166667em}{0ex}}i\ge 2\}$ and let ${A}_{2}$ be the ideal of R generated by $\overline{{X}_{1}}$. Then ${A}_{2}$ is a principal ideal of R; so ${A}_{2}$ is an S-finite ideal of R. Suppose to the contrary that ${A}_{1}$ is an S-finite ideal of R. Then there exist an element $s\in S$ and a finitely generated ideal F of R such that $s{A}_{1}\subseteq F\subseteq {A}_{1}$. Note that $F\subseteq (\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{2}},\cdots ,\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{m}})$ for some integer $m\ge 2$. However, an easy calculation shows that $s\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{m+1}}\notin (\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{2}},\cdots ,\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{m}})$; so $s\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{m+1}}\notin F$. This is a contradiction. Hence ${A}_{1}$ is not an S-finite ideal of R. Thus R is not a weakly S-Noetherian ring.
- (2)
- Let A be a proper ideal of R. Then for any $\overline{f}\in A$, there exists an element $g\in D[{X}_{1},{X}_{2}]$ such that $\overline{g}=\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{2}}\phantom{\rule{0.166667em}{0ex}}\overline{f}$. Let C be the ideal of $D[{X}_{1},{X}_{2}]$ generated by the set $\{g\in D[{X}_{1},{X}_{2}]\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\overline{g}\in \overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{2}}A\}$. Since $D[{X}_{1},{X}_{2}]$ is a Noetherian ring, we have that $C=({g}_{1},\cdots ,{g}_{n})$ for some ${g}_{1},\cdots ,{g}_{n}\in D[{X}_{1},{X}_{2}]$. Note that $\overline{{X}_{1}}\phantom{\rule{0.166667em}{0ex}}\overline{{X}_{2}}A=(\overline{{g}_{1}},\cdots ,\overline{{g}_{n}})$; so A is a T-finite ideal of R. Hence R is a T-Noetherian ring. Thus R is a weakly T-Noetherian ring.

**Proposition**

**2.**

**Proof**.

**Corollary**

**3.**

**Proof**.

**Corollary**

**4.**

**Proof**.

**Corollary**

**5.**

**Proof**.

**Proposition**

**3.**

**Proof**.

**Corollary**

**6.**

**Proof**.

**Corollary**

**7.**

- (1)
- R is a weakly S-Noetherian ring.
- (2)
- ${R}_{S}$ is a weakly Noetherian ring.
- (3)
- ${R}_{S}$ is a weakly $\psi \left(S\right)$-Noetherian ring.

**Proof**.

**Remark**

**1.**

**Example**

**3.**

- (1)
- Let $S=\{(1,p,p,0,0,\cdots ),(1,0,{p}^{2},0,0,\cdots ),(1,0,0,\cdots )\}$. Then S is a multiplicative subset of R. Let I be an ideal of R. Then for any $s\in S$, $sI$ is a finite ideal of R; so I is an S-finite ideal of R. Hence R is an S-Noetherian ring, and thus R is a weakly S-Noetherian ring.
- (2)
- Let I be the ideal of R generated by $(1,p,p,\cdots )$ and J the ideal of R generated by the set $\left\{\right(1,p,0,0,\cdots ),(1,0,p,0,0,\cdots ),(1,0,0,p,0,0,\cdots ),\cdots \}$. Then J is a subideal of the principal ideal I. However, J is not finitely generated. Thus R is not a weakly Noetherian ring.

**Example**

**4.**

- (1)
- Note that $\left(\overline{{X}_{1}}\right)\subseteq (\overline{{X}_{1}},\overline{{X}_{2}})\subseteq \cdots $ is a strictly ascending chain of ideals of R; so R is not a Noetherian ring. Since R contains a nonunit regular element $\overline{1+{X}_{1}}$, we have that R is not a weakly Noetherian ring [1] (Theorem 1(2)).
- (2)
- Let S be the multiplicative subset of R generated by $\overline{{X}_{1}}$ and let s be any element of S. Let B be a proper ideal of R. Then for any $\overline{f}\in B$, we can find an element $g\in D\left[{X}_{1}\right]$ such that $\overline{g}=s\overline{f}$. Let C be the ideal of $D\left[{X}_{1}\right]$ generated by the set $\{g\in D\left[{X}_{1}\right]\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\overline{g}\in sB\}$. Since $D\left[{X}_{1}\right]$ is a Noetherian ring, we have that $C=({g}_{1},\cdots ,{g}_{n})$ for some ${g}_{1},\cdots ,{g}_{n}\in D\left[{X}_{1}\right]$. Note that $sB=(\overline{{g}_{1}},\cdots ,\overline{{g}_{n}})$; so B is an S-finite ideal of R. Hence R is an S-Noetherian ring. Thus R is a weakly S-Noetherian ring.

**Proposition**

**4.**

- (1)
- R is a weakly Noetherian ring.
- (2)
- R is a weakly P-Noetherian ring for all prime ideals P of R.
- (3)
- R is a weakly M-Noetherian ring for all maximal ideals M of R.

**Proof**.

**Corollary**

**8.**

**Example**

**5.**

- (1)
- Let J be a finitely generated proper ideal of R and I a subideal of J. Then $J=(\overline{{f}_{1}},\cdots ,\overline{{f}_{m}})$ for some ${f}_{1},\cdots ,{f}_{m}\in F{\left[\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{X}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{1}$; so ${f}_{1},\cdots ,{f}_{m}\in F\left[\phantom{\rule{-0.166667em}{0ex}}[{X}_{1},\cdots ,{X}_{n}]\phantom{\rule{-0.166667em}{0ex}}\right]$ for some integer $n\ge 1$. Therefore $({f}_{1},\cdots ,{f}_{m})F\left[\phantom{\rule{-0.166667em}{0ex}}[{X}_{1},\cdots ,{X}_{n}]\phantom{\rule{-0.166667em}{0ex}}\right]\subseteq ({X}_{1},\cdots ,{X}_{n})F\left[\phantom{\rule{-0.166667em}{0ex}}[{X}_{1},\cdots ,{X}_{n}]\phantom{\rule{-0.166667em}{0ex}}\right]$, which shows that $({f}_{1},\cdots ,{f}_{m})F{\left[\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{X}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{1}\subseteq ({X}_{1},\cdots ,{X}_{n})F{\left[\phantom{\rule{-0.166667em}{0ex}}\left[\mathbf{X}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{1}$. Hence $J\subseteq (\overline{{X}_{1}},\cdots ,\overline{{X}_{n}})R$, which indicates that every element of I is of the form $\overline{{g}_{1}}\overline{{X}_{1}}+\cdots +\overline{{g}_{n}}\overline{{X}_{n}}$ for some ${g}_{i}\in F\left[\phantom{\rule{-0.166667em}{0ex}}\left[{X}_{i}\right]\phantom{\rule{-0.166667em}{0ex}}\right]$ for each $i\in \{1,\cdots ,n\}$. Let C be the ideal of $F\left[\phantom{\rule{-0.166667em}{0ex}}[{X}_{1},\cdots ,{X}_{n}]\phantom{\rule{-0.166667em}{0ex}}\right]$ generated by the set $\{{g}_{1}{X}_{1}+\cdots +{g}_{n}{X}_{n}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}$ for each $i=1,\cdots ,n$, ${g}_{i}\in F\left[\phantom{\rule{-0.166667em}{0ex}}\left[{X}_{i}\right]\phantom{\rule{-0.166667em}{0ex}}\right]$ and $\overline{{g}_{1}}\overline{{X}_{1}}+\cdots +\overline{{g}_{n}}\overline{{X}_{n}}\in I\}$. Since $F\left[\phantom{\rule{-0.166667em}{0ex}}[{X}_{1},\cdots ,{X}_{n}]\phantom{\rule{-0.166667em}{0ex}}\right]$ is a Noetherian ring, we have that $C=({h}_{1},\cdots ,{h}_{p})$ for some ${h}_{1},\cdots ,{h}_{p}\in F\left[\phantom{\rule{-0.166667em}{0ex}}[{X}_{1},\cdots ,{X}_{n}]\phantom{\rule{-0.166667em}{0ex}}\right]$; so $I=(\overline{{h}_{1}},\cdots ,\overline{{h}_{p}})$. Hence R is a weakly Noetherian ring. Thus by Remark 1, we have that R is a weakly S-Noetherian ring for any multiplicative subset S of R.
- (2)
- Let S be the set of units in R. Then S is a multiplicative subset of R. Note that $(\overline{{X}_{1}},\overline{{X}_{2}},\cdots )$ is not a finitely generated ideal of R; so R is not a Noetherian ring. Thus R is not an S-Noetherian ring.

**Proposition**

**5.**

- (1)
- R is a weakly S-Noetherian ring.
- (2)
- For all $i=1,\cdots ,n$, ${R}_{i}$ is an ${S}_{i}$-Noetherian ring.

**Proof**.

**Example**

**6.**

## 3. Some Extensions of Weakly S-Noetherian Rings

**Lemma**

**1.**

- (1)
- If R is a weakly S-Noetherian ring and I is an S-finite ideal of R, then $R/I$ is a weakly $(S/I)$-Noetherian ring.
- (2)
- If $R/I$ is a weakly $(S/I)$-Noetherian ring and I is an S-Noetherian R-module contained in $J\left(R\right)$, then R is a weakly S-Noetherian ring.

**Proof**.

**Example**

**7.**

- (1)
- $R/I$ is isomorphic to T; so $R/I$ is a weakly Noetherian ring which is not a Noetherian ring (cf. [1] (Example 1)). Hence $R/I$ is a weakly ($S/I$)-Noetherian ring by Remark 1.
- (2)
- Note that $J\left(R\right)=J\left(T\right)(+)T$; so $I\subseteq J\left(R\right)$.
- (3)
- Note that I is a finitely generated ideal of R; so I is an S-finite R-module.
- (4)
- R is not a weakly S-Noetherian ring because I is an S-finite R-module but $\left\{\right(0,\mathbf{O}\left)\right\}(+)M$ is not an S-finite R-module.

**Theorem**

**1.**

- (1)
- If $R(+)M$ is a weakly $\left(S\right(+\left)M\right)$-Noetherian ring, then R is a weakly S-Noetherian ring.
- (2)
- If R is a weakly S-Noetherian ring and M is an S-Noetherian R-module, then $R(+)M$ is a weakly $\left(S\right(+\left)M\right)$-Noetherian ring.

**Proof**.

**Corollary**

**9.**

- (1)
- R is an S-Noetherian ring and M is an S-Noetherian R-module.
- (2)
- $R(+)M$ is an $\left(S\right(+\left)M\right)$-Noetherian ring.

**Proof**.

**Example**

**8.**

- (1)
- R is a weakly Noetherian ring which is not a Noetherian ring [1] (Example 1). Thus by Remark 1, we have that R is a weakly S-Noetherian ring which is not an S-Noetherian ring.
- (2)
- Note that R is a finitely generated R-module; so R is an S-finite R-module. However, by (1), we have that R is not an S-Noetherian R-module.
- (3)
- Note that $\left\{\right(0,\mathbf{O}\left)\right\}(+)R$ is an ($S(+)R$)-finite ($R(+)R$)-module but $\left\{\right(0,\mathbf{O}\left)\right\}(+)N$ is not an ($S(+)R$)-finite ($R(+)R$)-module. Hence $R(+)R$ is not a weakly ($S(+)R$)-Noetherian ring.

**Proposition**

**6.**

- (1)
- M is an S-Noetherian R-module.
- (2)
- R is a weakly S-Noetherian ring if and only if $R(+)M$ is a weakly $\left(S\right(+\left)M\right)$-Noetherian ring.

**Proof**.

**Theorem**

**2.**

- (1)
- If $R{\bowtie}^{f}J$ is a weakly ${S}^{\prime}$-Noetherian ring and J is an S-finite R-module, then R is a weakly S-Noetherian ring.
- (2)
- If R is a weakly S-Noetherian ring and J is an S-Noetherian R-module contained in $J\left(T\right)$, then $R{\bowtie}^{f}J$ is a weakly ${S}^{\prime}$-Noetherian ring.

**Proof**.

**Corollary**

**10.**

- (1)
- If $R\bowtie I$ is a weakly ${S}^{\prime}$-Noetherian ring and I is an S-finite ideal of R, then R is a weakly S-Noetherian ring.
- (2)
- If R is a weakly S-Noetherian ring and I is an S-Noetherian R-module contained in $J\left(R\right)$, then $R\bowtie I$ is a weakly ${S}^{\prime}$-Noetherian ring.

**Question**

**1.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Kim, D.K.; Lim, J.W.
A Note on Weakly *S*-Noetherian Rings. *Symmetry* **2020**, *12*, 419.
https://doi.org/10.3390/sym12030419

**AMA Style**

Kim DK, Lim JW.
A Note on Weakly *S*-Noetherian Rings. *Symmetry*. 2020; 12(3):419.
https://doi.org/10.3390/sym12030419

**Chicago/Turabian Style**

Kim, Dong Kyu, and Jung Wook Lim.
2020. "A Note on Weakly *S*-Noetherian Rings" *Symmetry* 12, no. 3: 419.
https://doi.org/10.3390/sym12030419