# Fuzzy Normed Rings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

- 1.
- $\Vert a\Vert =0\iff a=0$,
- 2.
- $\Vert a+b\Vert \le \Vert a\Vert +\Vert b\Vert $,
- 3.
- $\Vert -a\Vert =\Vert a\Vert $, (and hence$\Vert {1}_{A}\Vert =1=\Vert -1\Vert $if identity exists), and
- 4.
- $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert $.

**Definition**

**3.**

- (1)
- $\ast $is commutative and associative, that is,$\ast (x,y)=\ast (y,x)$and$\ast (x,\ast (y,z))=\ast (\ast (x,y),z)$,
- (2)
- $\ast $is continuous,
- (3)
- $\ast (x,1)=x$,
- (4)
- $\ast $is monotone, which means$\ast (x,y)\le \ast (x,z)$if$y\le z$,
- (5)
- $\ast (x,x)x$for$x\in (0,1)$, and
- (6)
- when$x<z$and$y<t$,$\ast (x,y)\ast (z,t)$for all$x,y,z,t\in (0,1)$.

**Definition**

**4.**

- (1)
- $\diamond $is commutative and associative, that is,$\diamond (x,y)=\diamond (y,x)$and$\diamond (x,\diamond (y,z))=\diamond (\diamond (x,y),z)$,
- (2)
- $\diamond $is continuous,
- (3)
- $\diamond (x,0)=x$,
- (4)
- $\diamond $is monotone, which means$\diamond (x,y)\le \diamond (x,z)$if$y\le z$,
- (5)
- $\diamond (x,x)x$for$x\in (0,1)$, and
- (6)
- when$x<z$and$y<t$,$\diamond (z,t)\diamond (x,y)$for all$x,y,z,t\in (0,1)$.

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Note:**The fuzzy operations of the fuzzy subsets$A,B\in F(R)$on the ring$R$can be extended to the operations below by t-norms and s-norms:

## 3. Fuzzy Normed Rings and Fuzzy Normed Ideals

**Definition**

**9.**

- (i)
- $A(x-y)\ge A(x)\ast A(y)$
- (ii)
- $A(x.y)\ge A(x)\ast A(y)$.

**Example**

**1.**

**Lemma**

**1.**

**Proof.**

**□**

**Lemma**

**2.**

- i.
- Let$A$be a fuzzy normed subring of the normed ring$NR$and let$f:NR\to NR\prime $be a ring homomorphism. Then,$f(A)$is a fuzzy normed subring of$NR\prime $.
- ii.
- Let$f:NR\to NR\prime $be a normed ring homomorphism. If$B$is a fuzzy normed subring of$NR\prime $, then${f}^{-1}(B)$is a fuzzy normed subring of$NR$.

**Proof.**

**□**

**Definition**

**10.**

**Definition**

**11.**

- (i)
- $A(x-y)\ge A(x)\ast A(y)$and
- (ii)
- $A(x.y)\ge A(y)$$\left(A(x.y)\ge A(x)\right)$,

**Definition**

**12.**

- (i)
- $A(x-y)\ge A(x)\ast A(y)$and
- (ii)
- $A(x.y)\ge A(x)\diamond A(y)$,

**Remark**

**1.**

**Example**

**2.**

**Solution:**Let $x,y\in NR$.

**Example**

**3.**

**Theorem**

**1.**

- (i)
- $w\in FN(X)\Rightarrow A(w)\ge \underset{1\le i\le m}{\ast}\left(A({a}_{i})\right)$,
- (ii)
- $x\in (y)\Rightarrow A(x)\ge A(y)$,
- (iii)
- $A\left(0\right)\ge A(x)$and
- (iv)
- if 1 is the multiplicative identity of$NR$, then$A(x)\ge A\left(1\right)$.

**Proof.**

## 4. Fuzzy Normed Prime Ideal and Fuzzy Normed Maximal Ideal

**Definition**

**13.**

**Lemma**

**3.**

**Proof.**

**Definition**

**14.**

**Example**

**4.**

**Solution:**As $I\ne NR$, ${\lambda}_{I}$ is a non-constant function on $NR$. Let $A$ and $B$ be two fuzzy normed ideals on $NR$ such that $A\circ B\subseteq {\lambda}_{I}$, but $A\not\subset {\lambda}_{I}$ and $B\not\subset {\lambda}_{I}$. There exists $x,y\in NR$ such that $A(x)\overline{)\le}{\lambda}_{I}(x)$ and $B(y)\overline{)\le}{\lambda}_{I}(y)$. In this case, $A(x)\ne 0$ and $B(y)\ne 0$, but ${\lambda}_{I}(x)=0$ and ${\lambda}_{I}(y)=0$. Therefore $x\notin I,y\notin I$. As $I$ is a fuzzy normed prime ideal, there exists an $r\in NR$, such that $xry\notin I$. This is obvious, because if $I$ is fuzzy normed prime, $A\circ B(xry)\subseteq I\Rightarrow A(x)\subseteq I\mathrm{or}B(ry)\subseteq I$ and therefore as $(NRxNR)(\underset{\_}{NRr}yNR)=(NRxNR)(NRyNR)\subseteq I$, we have either $NRxNR\subseteq I$ or $NRyNR\subseteq I$. Assume $NRxNR\subseteq I$. Then $xxx={(x)}^{3}\in I\Rightarrow x\subseteq I$, but this contradicts with the fact that ${\lambda}_{I}(x)=0$. Now let $a=xry$. ${\lambda}_{I}(a)=0$. Thus, $A\circ B(a)=0$. On the other hand,

**Theorem**

**2.**

**Proof:**

**Definition**

**15.**

**Example**

**5.**

**Theorem**

**3.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Naĭmark, M.A. Normed Rings; Noordhoff: Groningen, The Netherlands, 1964. [Google Scholar]
- Arens, R. A generalization of normed rings. Pac. J. Math.
**1952**, 2, 455–471. [Google Scholar] [CrossRef] - Gel’fand, I.; Raikov, M.D.A.; Shilov, G.E. Commutative Normed Rings; Chelsea Publishing Company: New York, NY, USA, 1964. [Google Scholar]
- Jarden, M. Normed Rings-Algebraic Patching; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Zadeh, L.A. Fuzzy Sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Zimmermann, H.J. Fuzzy Set Theory and Its Application, 2nd ed.; Kluwer Academic Publishers: Norwell, MA, USA, 1991. [Google Scholar]
- Mordeson, J.N.; Bhutani, K.R.; Rosenfeld, A. Fuzzy Group Theory; Springer: Heidelberg, Germany, 2005. [Google Scholar]
- Liu, W. Fuzzy Invariant Subgroups and Fuzzy Ideals. Fuzzy Sets Syst.
**1982**, 8, 133–139. [Google Scholar] [CrossRef] - Mukherjee, N.P.; Bhattacharya, P. Fuzzy Normal Subgroups and Fuzzy Cosets. Inf. Sci.
**1984**, 34, 225–239. [Google Scholar] [CrossRef] - Wang, X.; Ruan, D.; Kerre, E.E. Mathematics of Fuzziness—Basic Issues; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Swamy, U.M.; Swamy, K.L.N. Fuzzy Prime Ideals of Rings. J. Math. Anal. Appl.
**1988**, 134, 94–103. [Google Scholar] [CrossRef] - Gupta, M.M.; Qi, J. Theory of T-norms and Fuzzy Inference Methods. Fuzzy Sets Syst.
**1991**, 40, 431–450. [Google Scholar] [CrossRef] - Kolmogorov, A.N.; Silverman, R.A.; Fomin, S.V. Introductory Real Analysis; Dover Publications, Inc.: New York, NY, USA, 1970. [Google Scholar]
- Uluçay, V.; Şahin, M.; Olgun, N. Normed Z-Modules. Int. J. Pure Appl. Math.
**2017**, 112, 425–435. [Google Scholar] - Şahin, M.; Olgun, N.; Uluçay, V. Soft Normed Rings. Springer Plus
**2016**, 5, 1950–1956. [Google Scholar] - Şahin, M.; Olgun, N.; Uluçay, V. Normed Quotient Rings. New Trends Math. Sci.
**2018**, 6, 52–58. [Google Scholar] - Şahin, M.; Kargın, A. Neutrosohic Triplet Normed Space. Open Phys.
**2017**, 15, 697–704. [Google Scholar] [CrossRef] - Olgun, N.; Şahin, M. Fitting ideals of universal modules. In Proceedings of the International Conference on Natural Science and Engineering (ICNASE’16), Kilis, Turkey, 19–20 March 2016; pp. 1–8. [Google Scholar]
- Olgun, N. A Problem on Universal Modules. Commun. Algebra
**2015**, 43, 4350–4358. [Google Scholar] [CrossRef] - Şahin, M.; Kargın, A. Neutrosophic Triplet Inner Product. Neutrosophic Oper. Res.
**2017**, 2, 193–205. [Google Scholar] - Smarandache, F.; Şahin, M.; Kargın, A. Neutrosophic Triplet G-Module. Mathematics
**2018**, 6, 53. [Google Scholar] [CrossRef] - Şahin, M.; Olgun, N.; Kargın, A.; Uluçay, V. Isomorphism theorems for soft G-modules. Afr. Mat.
**2018**, 1–8. [Google Scholar] [CrossRef] - Çelik, M.; Shalla, M.; Olgun, N. Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups. Symmetry
**2018**, 10, 321–335. [Google Scholar] - Bal, M.; Shalla, M.M.; Olgun, N. Neutrosophic Triplet Cosets and Quotient Groups. Symmetry
**2018**, 10, 126. [Google Scholar] [CrossRef]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Emniyet, A.; Şahin, M.
Fuzzy Normed Rings. *Symmetry* **2018**, *10*, 515.
https://doi.org/10.3390/sym10100515

**AMA Style**

Emniyet A, Şahin M.
Fuzzy Normed Rings. *Symmetry*. 2018; 10(10):515.
https://doi.org/10.3390/sym10100515

**Chicago/Turabian Style**

Emniyet, Aykut, and Memet Şahin.
2018. "Fuzzy Normed Rings" *Symmetry* 10, no. 10: 515.
https://doi.org/10.3390/sym10100515