The normalizer problem for finite groups having normal $2$-complements

1.   Introduction

The notion of intuitionistic fuzzy normed subring and intuitionistic fuzzy normed ideal was characterized by Abed Alhaleem and Ahmad in ^[10], after that the necessity has arisen to introduce the concepts of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals. Following the work of Emniyent and Şahin in ^[17] which outlined the concepts of fuzzy normed prime ideal and maximal ideal we implement the conception of intuitionistic fuzzy to prime and maximal normed ideals. After the establishment of fuzzy set by Zadeh ^[28] which showed that the membership of an element in a fuzzy set is at intervals [0, 1], many researchers investigated on the properties of fuzzy set because it handles uncertainty and vagueness, and due to its applications in many fields of studies. A lot of work has been done on various aspects and for the last 50 years, the relation betwee maximal and prime ideals has become the core of many researchers work. Swamy and Swamy in 1988 ^[27] presented the conceptions of fuzzy ideal and fuzzy prime ideal with truth values in a complete lattice fulfilling the infinite distributive law. Later, many researchers studied the generalization of fuzzy ideals and fuzzy prime (maximal) ideals of rings: Dixit et al ^[16], Malik and Mordeson in ^[22] and Mukherjee and Sen in ^[24]. The notion of intuitionistic fuzzy set was initiated by Atanassov ^[6], as a characterization of fuzzy set which assigned the degree of membership and the degree of non-membership for set elements, he also delineated some operations and connections over basic intuitionistic fuzzy sets. In ^[5], Atanassov introduced essential definitions and properties of the interval-valued intuitionistic fuzzy sets and the explanation of mostly extended modal operator through interval-valued intuitionistic fuzzy sets were presented in ^[4], and some of its main properties were studied. Banerjee and Basnet ^[13] investigated intuitionistic fuzzy rings and intuitionistic fuzzy ideals using intuitionistic fuzzy sets. In 2005 ^[20], an identification of intuitionistic fuzzy ideals, intuitionistic fuzzy prime ideals and intuitionistic fuzzy completely prime ideals was given. In ^[14], Bakhadach et al. implemented the terms of intuitionistic fuzzy ideals and intuitionistic fuzzy prime (maximal) ideals, investigated these notions to show new results using the intuitionistic fuzzy points and membership and nonmembership functions. The paper comprises the following: we begin with the preliminary section, we submit necessary notations and elementary outcomes. In Section 3, we characterize some properties of intuitionistic fuzzy normed ideals and identify the image and the inverse image of intuitionistic fuzzy normed ideals. In Section 4, we describe the notions of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals and we characterize the relation between the intuitionistic characteristic function and prime (maximal) ideals. In Section 5, the conclusions are outlined.

2.   Preliminaries

We first include some definitions needed for the subsequent sections:

Definition 2.1. ^[25] A linear space $L$ is called a normed space if for any element $r$ there is a real number $\Vert r \Vert$ satisfying:

$\bullet \quad \Vert r \Vert \geq 0$ for every $r \in L$, when $r = 0$ then $\Vert r \Vert = 0$;

$\bullet \quad \Vert \alpha.r \Vert = \vert \alpha \vert.\Vert r \Vert$;

$\bullet \quad \Vert r+v \Vert \leq \Vert r \Vert + \Vert v \Vert$ for all $r, v \in L$.

Definition 2.2. ^[18] A ring $R$ is said to be a normed ring $(NR)$ if it possesses a norm $\Vert \; \Vert$, that is, a non-negative real-valued function $\Vert \; \Vert : NR \rightarrow \mathbb{R}$ such that for any $r, v \in R$,

$1) \; \Vert r \Vert = 0 \Leftrightarrow r = 0$,

$2) \; \Vert r+v \Vert \leq \Vert r \Vert + \Vert v \Vert$,

$3) \; \Vert r \Vert = \Vert -r \Vert$, (and hence $\Vert 1_A \Vert = 1 = \Vert -1 \Vert$ if identity exists), and

$4) \; \Vert rv \Vert \leq \Vert r \Vert \; \Vert v \Vert$.

Definition 2.3. ^[1] Let $\ast:[0, 1] \times [0, 1]\rightarrow [0, 1]$ be a binary operation. Then $\ast$ is a $t$-norm if $\ast$ conciliates the conditions of commutativity, associativity, monotonicity and neutral element 1.

We shortly use $t$-norm and write $r \ast v$ instead of $\ast(r, v)$.

Two examples of continuous $t$-norm are: $r \ast v = rv$ and $r \ast v = min\lbrace r, v \rbrace$ ^[26].

Proposition 2.4. ^[21] A $t$-norm $T$ has the property, for every $r, v \in [0, 1]$

Definition 2.5. ^[19] Let $\diamond:[0, 1]\times [0, 1]\rightarrow [0, 1]$ be a binary operation. Then $\diamond$ is a $s$-norm if $\diamond$ conciliates the conditions of commutativity, associativity, monotonicity and neutral element 0.

We shortly use $s$-norm and write $r\diamond v$ instead of $\diamond(r, v)$.

Two examples of continuous $s$-norm are: $r \diamond v = min (r+v, 1)$ and $r \diamond v = max\lbrace r, v \rbrace$ ^[26].

Proposition 2.6. ^[21] A $s$-norm $S$ has the property, for every $r, v \in [0, 1]$

Definition 2.7. ^[28] A membership function $\mu_{A}(r):X \rightarrow [0, 1]$ specifies the fuzzy set $A$ over $X$, where $\mu_{A}(r)$ defines the membership of an element $r \in X$ in a fuzzy set $A$.

Definition 2.8. ^[6] An intuitionistic fuzzy set $A$ in set $X$ is in the form $IFS \; A = \lbrace(r, \mu_{A}(r), \gamma_{A}(r):r \in X\rbrace$, such that the degree of membership is $\mu_{A}(r):X \rightarrow [0, 1]$ and the degree of non-membership is $\gamma_{A}(r):X \rightarrow [0, 1]$, where $0 \leq \mu_{A}(r)+ \gamma_{A}(r))\leq 1$ for all $r \in X$. We shortly use $A = (\mu_{A}, \gamma_{A})$.

Definition 2.9. ^[7] Let $A$ be an intuitionistic fuzzy set in a ring $R$, we indicate the $(\alpha, \beta)$-cut set by $A_{\alpha, \beta} = \lbrace r\in R:\mu_{A}\geq\alpha$ and $\gamma_{A}\leq\beta\rbrace$ such that $\alpha+\beta\leq 1$ and $\alpha, \beta\in [0, 1]$.

Definition 2.10. ^[23] The support of an intuitionistic fuzzy set A, is denoted by $A_{\circ}$ and defined as $A_{\circ} = \lbrace r:\mu_{A}(r) > 0$ and $\gamma_{A}(r) < 1\rbrace$.

Definition 2.11. ^[2] The complement, union and intersection of two $IFS \; A = (\mu_{A}, \gamma_{A})$ and $B = (\mu_{B}, \gamma_{B})$, in a ring $R$, are defined as follows:

$1) \; A^{c} = \lbrace \langle r, \gamma_{A}(r), \mu_{A}(r)\rangle:r \in R\rbrace$,

$2) \; A \cup B = \lbrace \langle r, max(\mu_{A}(r), \mu_{B}(r)), min(\gamma_{A}(r), \gamma_{B}(r))\rangle:r \in R\rbrace$,

$3) \; A \cap B = \lbrace \langle r, min(\mu_{A}(r), \mu_{B}(r)), max(\gamma_{A}(r), \gamma_{B}(r))\rangle:r \in R\rbrace$.

Definition 2.12. ^[12] Let $NR$ be a normed ring. Then an IFS $A = \lbrace(r, \mu_{A}(r), \gamma_{A}(r)):r \in NR\rbrace$ of $NR$ is an intuitionistic fuzzy normed subring ($IFNSR$) of $NR$ if:

i. $\mu_A (r-v)\geq \mu_A (r) \ast \mu_A (v)$,

ii. $\mu_A (rv) \geq \mu_A (r)\ast \mu_A (v)$,

iii. $\gamma_A (r-v)\leq \gamma_A (r)\diamond \gamma_A (v)$,

iv. $\gamma_A (rv)\leq \gamma_A (r) \diamond \gamma_A (v)$.

Definition 2.13. ^[9] Let $NR$ be a normed ring. Then an IFS $A = \lbrace(r, \mu_{A}(r), \gamma_{A}(r)):r \in NR\rbrace$ of $NR$ is an intuitionistic fuzzy normed ideal ($IFNI$) of $NR$ if:

i. $\mu_{A}(r -v) \geq \mu_{A}(r) \ast \mu_{A}(v)$,

ii. $\mu_{A}(rv)\geq \mu_{A}(r)\diamond \mu_{A}(v)$,

iii. $\gamma_{A}(r-v) \leq \gamma_{A}(r)\diamond \gamma_{A}(v))$,

iv. $\gamma_{A}(rv)) \leq \gamma_{A}(r) \ast \gamma_{A}(v)\rbrace$.

Definition 2.14. ^[3] If $A$ and $B$ are two fuzzy subsets of the normed ring $NR$. Then the product $A\circ B(r)$ is defined by:

Definition 2.15. ^[22] A fuzzy ideal $A$ (non-constant) of a ring $R$ is considered to be a fuzzy prime ideal if $B \circ C \subseteq A$ for a fuzzy ideals $B$, $C$ of $R$ indicates that either $B \subseteq A$ or $C \subseteq A$.

3.   Properties of intuitionistic fuzzy normed ideal

In this section, we characterize several properties of intuitionistic fuzzy normed ideals and elementary results are obtained.

Definition 3.1. ^[8] Let $A$ and $B$ be two intuitionistic fuzzy subsets of the normed ring $NR$. The operations are defined as:

and

Therefore, the intrinsic product of $A$ and $B$ is considered to be the intuitionistic fuzzy normed set $A \circ B = (\mu_{A \circledast B}, \gamma_{A \otimes B}) = (\mu_{A}\circledast \mu_{B}, \gamma_{A}\otimes \gamma_{B})$.

Theorem 3.2. ^[10] Let $A$ and $B$ be two intuitionistic fuzzy ideals of a normed ring $NR$. Then $A\cap B$ is an intuitionistic fuzzy normed ideal of $NR$.

Example 3.1. Let $NR = \mathbb{Z}$ the ring of integers under ordinary addition and multiplication of integers.

Define the intuitionistic fuzzy normed subsets as $A = (\mu_{A}, \gamma_{A})$ and $B = (\mu_{B}, \gamma_{B})$, by

As $\mu_{A\cap B}(r) = min \lbrace\mu_{A}(r), \mu_{B}(r)\rbrace$ and $\gamma_{A\cap B}(r) = max \lbrace\gamma_{A}(r), \gamma_{B}(r)\rbrace$. Then,

It can be verified that $A$, $B$ and $A\cap B$ are intuitionistic fuzzy normed ideals of $NR$.

Lemma 3.3. Let $A$ and $B$ be an intuitionistic fuzzy normed right ideal and an intuitionistic fuzzy normed left ideal of a normed ring $NR$, respectively, then $A \circ B \subseteq A \cap B$ i.e, $A \circledast B (r)\leq A\cap B(r)\leq A \otimes B(r)$, where

Proof. Let $A\cap B$ be an intuitionistic fuzzy normed ideal of $NR$. Assume that $A$ is an intuitionistic fuzzy normed right ideal and $B$ is an intuitionistic fuzzy normed left ideal. Let $\mu_{A \circledast B}(r) = { \diamond \atop r = vz}(\mu_{A}(v)\ast \mu_{B}(z))$ and let $\gamma_{A \otimes B}(r) = { \ast \atop r = vz}(\gamma_{A}(v)\diamond \gamma_{B}(z))$.

Since, $A$ is an intuitionistic fuzzy normed right ideal and $B$ is an intuitionistic fuzzy normed left ideal, we have

and

Thus,

and

By (3.1) and (3.2) the proof is concluded.

Remark 3.4. The union of two intuitionistic fuzzy normed ideals of a ring $NR$ needs not be always intuitionistic fuzzy normed ideal.

Example 3.2. Let $NR = \mathbb{Z}$ the ring of integers under ordinary addition and multiplication of integers.

Let the intuitionistic fuzzy normed subsets $A = (\mu_{A}, \gamma_{A})$ and $B = (\mu_{B}, \gamma_{B})$, define by

It can be checked that $A$ and $B$ are intuitionistic fuzzy normed ideals of $NR$.

As $\mu_{A\cup B}(r) = max \lbrace\mu_{A}(r), \mu_{B}(r)\rbrace$ and $\gamma_{A\cup B}(r) = min \lbrace\gamma_{A}(r), \gamma_{B}(r)\rbrace$. Then,

Let $r = 15$ and $v = 4$, then $\mu_{A\cup B}(15) = 0.85$, $\mu_{A\cup B}(4) = 0.75$ and $\gamma_{A \cup B}(15) = 0.2$, $\gamma_{A \cup B}(4) = 0.3$.

Hence, $\mu_{A\cup B}(15-4) = \mu_{A\cup B}(11) = 0.35 \ngtr \mu_{A\cup B}(15) \ast \mu_{A\cup B}(4) = min \lbrace 0.85, 0.75 \rbrace$ and $\gamma_{A \cup B}(15-4) = \gamma_{A \cup B}(11) = 0.4 \nless \gamma_{A \cup B}(15) \diamond \gamma_{A \cup B}(4) = max \lbrace 0.2, 0.3 \rbrace$. Thus, the union of two intuitionistic fuzzy normed ideals of $NR$ need not be an intuitionistic fuzzy normed ideal.

Proposition 3.5. Let $A = (\mu_A, \gamma_A)$ be an intuitionistic fuzzy normed ideal of a ring $NR$, then we have for all $r \in NR$:

i. $\mu_A (0) \geq \mu_A (r)$ and $\gamma_A (0) \leq \gamma_A (r)$,

ii. $\mu_A (-r) = \mu_A (r)$ and $\gamma_A (-r) = \gamma_A (r)$,

iii. If $\mu_A (r-v) = \mu_A (0)$ then $\mu_A (r) = \mu_A (v)$,

iv. If $\gamma _A (r-v) = \gamma _A (0)$ then $\gamma _A (r) = \gamma _A (v)$.

Proof. i. As $A$ is an intuitionistic fuzzy normed ideal, then

and

ii. $\mu_A (-r) = \mu_A (0 -r) \geq \mu_A (0) \ast \mu_A (r) = \mu_A (r)$ and $\mu_A (r) = \mu_A (0 - (-r)) \geq \mu_A (0) \ast \mu_A (-r) = \mu_A (-r)$.

Therefore, $\mu_A (-r) = \mu_A (r)$

also,

$\gamma_A (-r) = \gamma_A (0 -r) \leq \gamma_A (0) \diamond \gamma_A (r) = \gamma_A (r)$ and $\gamma_A (r) = \gamma_A (0 - (-r)) \leq \gamma_A (0) \diamond \gamma_A (-r) = \gamma_A (-r)$.

Therefore, $\gamma_A (-r) = \gamma_A (r)$.

iii. Since $\mu_A (r-v) = \mu_A (0)$, then

similarly

Consequently, $\mu_A (r) = \mu_A (v)$.

iv. same as in iii.

Proposition 3.6. Let $A$ be an intuitionistic fuzzy normed ideal of a normed ring $NR$, then $\bigtriangleup A = (\mu_{A}, \mu^{c}_{A})$ is an intuitionistic fuzzy normed ideal of $NR$.

Proof. Let $r, v \in NR$

Then $\mu^{c}_{A}(r-v) \leq \mu^{c}_{A}(r)\diamond \mu^{c}_{A}(v)$.

Then $\mu^{c}_{A}(rv) \leq \mu^{c}_{A}(r) \ast \mu^{c}_{A}(v)$.

Accordingly, $\bigtriangleup A = (\mu_{A}, \mu^{c}_{A})$ is an intuitionistic fuzzy normed ideal of $NR$.

Proposition 3.7. If $A$ is an intuitionistic fuzzy normed ideal of a normed ring $NR$, then $\lozenge A = (\gamma^{c}_{A}, \gamma_{A})$ is an intuitionistic fuzzy normed ideal of $NR$.

Proof. Let $r, v \in NR$

Then $\gamma^{c}_{A}(r-v) \geq \gamma^{c}_{A}(r)\ast \gamma^{c}_{A}(v)$.

Then $\gamma^{c}_{A}(rv) \geq \gamma^{c}_{A}(r) \diamond \gamma^{c}_{A}(v)$.

Therefore, $\lozenge A = (\gamma^{c}_{A}, \gamma_{A})$ is an intuitionistic fuzzy normed ideal of $NR$.

Proposition 3.8. An $IFS \; A = (\mu_{A}, \gamma_{A})$ is an intuitionistic fuzzy normed ideal of $NR$ if the fuzzy subsets $\mu_{A}$ and $\gamma^{c}_{A}$ are intuitionistic fuzzy normed ideals of $NR$.

Proof. Let $r, v \in NR$

Then, $\gamma_{A}(r - v) \leq \gamma_{A}(r) \diamond \gamma_{A}(v)$.

Then, $\gamma_{A}(rv) \leq \gamma_{A}(r) \ast \gamma_{A}(v)$.

Consequently, $A = (\mu_{A}, \gamma_{A})$ is an intuitionistic fuzzy normed ideal of $NR$.

Definition 3.9. Let $A$ be a set (non-empty) of the normed ring $NR$, the intuitionistic characteristic function of $A$ is defined as $\lambda_{A} = (\mu_{\lambda_{A}}, \gamma_{\lambda_{A}})$, where

Lemma 3.10. Let $A$ and $B$ be intuitionistic fuzzy sets of a normed ring $NR$, then:

(i) $\lambda_{A} \cap \lambda_{B} = \lambda_{A \cap B}$ (ii) $\lambda_{A} \circ \lambda_{B} = \lambda_{A \circ B}$ (iii) If $A \subseteq B$, then $\lambda_{A} \subseteq \lambda_{B}$

Theorem 3.11. For a non-empty subset $A$ of $NR$, $A$ is a subring of $NR$ if and only if $\lambda_{A} = (\mu_{\lambda_{A}}, \gamma_{\lambda_{A}})$ is an intuitionistic fuzzy normed subring of $NR$.

Proof. Suppose $A$ to be a subring of $NR$ and let $r, v \in NR$. If $r, v \in A$, then by the intuitionistic characteristic function properties $\mu_{\lambda_{A}}(r) = 1 = \mu_{\lambda_{A}}(v)$ and $\gamma_{\lambda_{A}}(r) = 0 = \gamma_{\lambda_{A}}(v)$. As $A$ is a subring, then $r-v$ and $rv \in A$. Thus, $\mu_{\lambda_{A}}(r-v) = 1 = 1 \ast 1 = \mu_{\lambda_{A}}(r) \ast \mu_{\lambda_{A}}(v)$ and $\mu_{\lambda_{A}}(rv) = 1 = 1 \ast 1 = \mu_{\lambda_{A}}(r) \ast \mu_{\lambda_{A}}(v)$, also $\gamma_{\lambda_{A}}(r-v) = 0 = 0 \diamond 0 = \gamma_{\lambda_{A}}(r) \diamond \gamma_{\lambda_{A}}(v)$ and $\gamma_{\lambda_{A}}(rv) = 0 = 0 \diamond 0 = \gamma_{\lambda_{A}}(r) \diamond \gamma_{\lambda_{A}}(v)$. This implies,

Similarly we can prove the above expressions if $r, v \notin A$.

Hence, $\lambda_{A} = (\mu_{\lambda_{A}}, \gamma_{\lambda_{A}})$ is an intuitionistic fuzzy normed subring of $NR$.

Conversely, we hypothesise that the intuitionistic characteristic function $\lambda_{A} = (\mu_{\lambda_{A}}, \gamma_{\lambda_{A}})$ is an intuitionistic fuzzy normed subring of $NR$. Let $r, v \in A$, then $\mu_{\lambda_{A}}(r) = 1 = \mu_{\lambda_{A}}(v)$ and $\gamma_{\lambda_{A}}(r) = 0 = \gamma_{\lambda_{A}}(v)$. So,

then $\mu_{\lambda_{A}}(r-v) = 1$, $\mu_{\lambda_{A}}(rv) = 1$ and $\gamma_{\lambda_{A}}(r-v) = 0$, $\gamma_{\lambda_{A}}(rv) = 0$, which implies that $r-v$ and $rv \in A$. Therefore, $A$ is a subring of $NR$.

Theorem 3.12. Let $I$ be a non-empty subset of a normed ring $NR$, then $I$ is an ideal of $NR$ if and only if $\lambda_{I} = (\mu_{\lambda_{I}}, \gamma_{\lambda_{I}})$ is an intuitionistic fuzzy normed ideal of $NR$.

Proof. Let $I$ be an ideal of $NR$ and let $r, v \in NR$.

Case I. If $r, v \in I$ then $rv \in I$ and $\mu_{\lambda_{I}}(r) = 1$, $\mu_{\lambda_{I}}(v) = 1$ and $\gamma_{\lambda_{I}}(r) = 0$, $\gamma_{\lambda_{I}}(v) = 0$. Thus, $\mu_{\lambda_{I}}(rv) = 1$ and $\gamma_{\lambda_{I}}(rv) = 0$. Accordingly, $\mu_{\lambda_{I}}(rv) = 1 = \mu_{\lambda_{I}}(r) \diamond \mu_{\lambda_{I}}(v)$ and $\gamma_{\lambda_{I}}(rv) = 0 = \gamma_{\lambda_{I}}(r) \ast \gamma_{\lambda_{I}}(v)$.

Case II. If $r \notin I$ or $v\notin I$ so $rv \notin I$, then $\mu_{\lambda_{I}}(r) = 0$ or $\mu_{\lambda_{I}}(v) = 0$ and $\gamma_{\lambda_{I}}(r) = 1$ or $\gamma_{\lambda_{I}}(v) = 1$. So, $\mu_{\lambda_{I}}(rv) = 1 \geq \mu_{\lambda_{I}}(r) \diamond \mu_{\lambda_{I}}(v)$ and $\gamma_{\lambda_{I}}(rv) = 0 \leq \gamma_{\lambda_{I}}(r) \ast \gamma_{\lambda_{I}}(v)$. Hence, $\lambda_{I} = (\mu_{\lambda_{I}}, \gamma_{\lambda_{I}})$ is an intuitionistic fuzzy normed ideal of $NR$.

On the hand, we suppose $\lambda_{I} = (\mu_{\lambda_{I}}, \gamma_{\lambda_{I}})$ is an intuitionistic fuzzy normed ideal of $NR$. The proof is similar to the second part of the proof of Theorem 3.11.

Proposition 3.13. If $A$ is an intuitionistic fuzzy normed ideal of $NR$, then $A_{\ast}$ is an ideal of $NR$ where $A_{\ast}$ is defined as,

Proof. See ^[10] (p. 6)

Lemma 3.14. Let $A$ and $B$ be two intuitionistic fuzzy normed left (right) ideal of $NR$. Therefore, $A_{\ast} \cap B_{\ast} \subseteq (A \cap B)_{\ast}$.

Proof. Let $r \in A_{\ast} \cap B_{\ast}$, then $\mu_{A}(r) = \mu_{A}(0)$, $\mu_{B}(r) = \mu_{B}(0)$ and $\gamma_{A}(r) = \gamma_{A}(0)$, $\gamma_{B}(r) = \gamma_{B}(0)$.

and

So, $r \in (A \cap B)_{\ast}$. Thus, $A_{\ast} \cap B_{\ast} \subseteq (A \cap B)_{\ast}$.

Theorem 3.15. Let $f:NR\rightarrow NR^{'}$ be an epimorphism mapping of normed rings. If $A$ is an intuitionistic fuzzy normed ideal of the normed ring $NR$, then $f(A)$ is also an intuitionistic fuzzy normed ideal of $NR^{'}$.

Proof. Suppose $A = \lbrace(r, \mu_{A}(r), \gamma_{A}(r)): r\in NR\rbrace$,

$f(A) = \lbrace(v, {\diamond \atop f(r) = v}\quad\mu_A (r), {\ast \atop f(r) = v}\quad\gamma_A (r): r\in NR, v\in NR^{'}\rbrace$.

Let $v_{1}, v_{2}\in NR^{'}$, then there exists $r_{1}, r_{2}\in NR$ such that $f(r_{1}) = v_{1}$ and $f(r_{2}) = v_{2}$.

i.

ii.

iii.

iv.

Hence, $f(A)$ is an intuitionistic fuzzy normed left ideal. Similarly, it can be justified that $f(A)$ is an intuitionistic fuzzy normed right ideal. Then, $f(A)$ is a intuitionistic fuzzy normed ideal of $NR^{'}$.

Proposition 3.16. Define $f:NR\rightarrow NR^{'}$ to be an epimorphism mapping. If $B$ is an intuitionistic fuzzy normed ideal of the normed ring $NR^{'}$, then $f^{-1}(B)$ is also an intuitionistic fuzzy normed ideal of $NR$.

Proof. Suppose $B = \lbrace(v, \mu_{B}(v), \gamma_{B}(v)):v \in NR^{'}\rbrace$, $f^{-1}(B) = \lbrace(r, \mu_{f^{-1}(B)}(r), \gamma_{f^{-1}(B)}(r): r\in NR\rbrace$, where $\mu_{f^{-1}(B)}(r) = \mu_{B}(f(r))$ and $\gamma_{f^{-1}(B)}(r) = \gamma_{B}(f(r))$ for every $r \in NR$. Let $r_{1}, r_{2}\in NR$, then

i.

ii.

iii.

iv.

Therefore, $f^{-1}(B)$ is an intuitionistic fuzzy normed left ideal of $NR$. Similarly, it can be justified that $f^{-1}(B)$ is an intuitionistic fuzzy normed right ideal. So, $f^{-1}(B)$ is a intuitionistic fuzzy normed ideal of $NR$.

4.   Intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal

In what follows, we produce the terms of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals and we investigate some associated properties.

Definition 4.1. An intuitionistic fuzzy normed ideal $A = (\mu_{A}, \gamma_{A})$ of a normed ring $NR$ is said to be an intuitionistic fuzzy normed prime ideal of $NR$ if for an intuitionistic fuzzy normed ideals $B = (\mu_{B}, \gamma_{B})$ and $C = (\mu_{C}, \gamma_{C})$ of $NR$ where $B\circ C \subseteq A$ indicates that either $B \subseteq A$ or $C \subseteq A$, which imply that $\mu_{B}\subseteq \mu_{A}$ and $\gamma_{A}\subseteq \gamma_{B}$ or $\mu_{C}\subseteq \mu_{A}$ and $\gamma_{A}\subseteq \gamma_{C}$.

Proposition 4.2. An intuitionistic fuzzy normed ideal $A = (\mu_{A}, \gamma_{A})$ is an intuitionistic fuzzy normed prime ideal if for any two intuitionistic fuzzy normed ideals $B = (\mu_{B}, \gamma_{B})$ and $C = (\mu_{C}, \gamma_{C})$ of $NR$ satisfies:

i. $\mu_{A} \supseteq \mu_{B \circledast C}$ i.e. $\mu_{A}(r)\geq { \diamond \atop r = vz}(\mu_{A}(v)\ast \mu_{B}(z))$;

ii. $\gamma_{A} \subseteq \gamma_{B \otimes C}$ i.e.$\gamma_{A}(r) \leq { \ast \atop r = vz}(\gamma_{A}(v)\diamond \gamma_{B}(z))$.

Theorem 4.3. Let $A$ be an intuitionistic fuzzy normed prime ideal of $NR$. Then $\mid$Im $\mu_{A}\mid$ = $\mid$Im $\gamma_{A}\mid = 2$; in other words $A$ is two-valued.

Proof. As $A$ is not constant, $\mid$Im $\mu_{A}\mid \geq 2$. assume that $\mid$Im $\mu_{A}\mid \geq 3$. $A_{\alpha, \beta} = \lbrace r\in R:\mu_{A}\geq\alpha$ and $\gamma_{A}\leq\beta\rbrace$ where $\alpha+\beta\leq 1$. Let $r \in NR$ and let $B$ and $C$ be two intuitionistic fuzzy subsets in $NR$, such that: $\mu_{A}(0) = s$ and $k = glb\lbrace \mu_{A}(r):r \in NR\rbrace$, so there exists $t, \alpha \in$ Im$(\mu_{A})$ such that $k \leq t < \alpha < s$ with $\mu_{B}(r) = \frac{1}{2}(t+\alpha)$, $\mu_{C}(r) = \left\{ \begin{array}{c l} s & , \; if \; r \in A_{\alpha, \beta} \\ k & , \; if \; r \notin A_{\alpha, \beta} \end{array}\right.$ and $\; \gamma_{A}(0) = c$ and $h = lub\lbrace \gamma_{A}(r):r \in NR\rbrace$, then there exists $d, \beta \in$ Im$(\gamma_{A})$ such that $c < \beta < d \leq h$ with $\gamma_{B}(r) = \frac{1}{2}(d+\beta)$ and $\gamma_{C}(r) = \left\{ \begin{array}{c l} c & , \; if \; r \; \in A_{\alpha, \beta} \\ h & , \; if \; r \notin A_{\alpha, \beta} \end{array}\right.$ for all $r \in NR$. Clearly $B$ is an intuitionistic fuzzy normed ideal of $NR$. Now we claim that $C$ is an intuitionistic fuzzy normed ideal of $NR$.

Let $r, v \in NR$, if $r, v\in A_{\alpha, \beta}$ then $r-v \in A_{\alpha, \beta}$ and $\mu_{C}(r-v) = s = \mu_{C}(r) \ast \mu_{C}(v)$, $\gamma_{C}(r-v) = c = \gamma_{C}(r) \diamond \gamma_{C}(v)$. If $r \in A_{\alpha, \beta}$ and $v \notin A_{\alpha, \beta}$ then $r-v \notin A_{\alpha, \beta}$ so, $\mu_{C}(r-v) = k = \mu_{C}(r) \ast \mu_{C}(v)$, $\gamma_{C}(r-v) = h = \gamma_{C}(r) \diamond \gamma_{C}(v)$. If $r, v \notin A_{\alpha, \beta}$ then $r-v \notin A_{\alpha, \beta}$ so, $\mu_{C}(r-v)\geq k = \mu_{C}(r) \ast \mu_{C}(v)$, $\gamma_{C}(r-v)\leq h = \gamma_{C}(r) \diamond \gamma_{C}(v)$. Hence, $\mu_{C}(r-v)\geq \mu_{C}(r) \ast \mu_{C}(v)$ and $\gamma_{C}(r-v)\leq \gamma_{C}(r) \diamond \gamma_{C}(v)$ for all $r, v \in NR$.

Now if $r \in A_{\alpha, \beta}$ then $rv \in A_{\alpha, \beta}$, thus $\mu_{C}(rv) = s = \mu_{C}(r) \diamond \mu_{C}(v)$ and $\gamma_{C}(rv) = c = \gamma_{C}(r) \ast \gamma_{C}(v)$. If $r \notin A_{\alpha, \beta}$, then $\mu_{C}(rv)\geq k = \mu_{C}(r) \diamond \mu_{C}(v)$ and $\gamma_{C}(rv)\leq h = \gamma_{C}(r) \ast \gamma_{C}(v)$. Therefore $C$ is an intuitionistic fuzzy normed ideal of $NR$.

To prove that $B\circ C \subseteq A$. Let $r \in NR$, we discuss the following cases:

(i) If $r = 0$, consequently

(ii) If $r \neq 0$, $r \in A_{\alpha, \beta}$. Then $\mu_{A}(r)\geq \alpha$ and $\gamma_{A}(r)\leq \beta$. Thus,

Since $\mu_{B}(u) \ast \mu_{C}(v) \leq \mu_{B}(u)$ and $\gamma_{B}(u) \diamond \gamma_{C}(v) \geq \gamma_{B}(u)$.

(iii) If $r \neq 0$, $r \notin A_{\alpha, \beta}$. Then in that case $u, v \in NR$ such that $r = uv$, $u \notin A_{\alpha, \beta}$ and $v \notin A_{\alpha, \beta}$. Then,

Therefore, in any case $\mu_{B \circledast C}(r) \leq \mu_{A}(r)$ and $\gamma_{B \otimes C}(r) \geq \gamma_{A}(r)$ for all $r \in NR$. Hence, $B\circ C \subseteq A$.

Let $a, b \in NR$ such that $\mu_{A}(a) = t$, $\mu_{A}(b) = \alpha$ and $\gamma_{A}(a) = d$, $\gamma_{A}(b) = \beta$. Thus, $\mu_{B}(a) = \frac{1}{2}(t+\alpha) > t = \mu_{A}(r)$ and $\gamma_{B}(a) = \frac{1}{2}(d+\beta) < d = \gamma_{A}(r)$ which implies that $B \nsubseteq A$. Also, $\mu_{A}(b) = \alpha$ and $\gamma_{A}(b) = \beta$ imply that $b \in A_{\alpha, \beta}$ so, $\mu_{C}(b) = s > \alpha$ and $\gamma_{C}(b) = c < \beta$, so $C \nsubseteq A$. Therefore, neither $B \nsubseteq A$ nor $C \nsubseteq A$. This indicates that $A$ could not be an intuitionistic fuzzy normed prime ideal of $NR$, so its a contradiction. Thus, $\mid$Im $\mu_{A}\mid$ = $\mid$Im$\gamma_{A}\mid = 2$.

Proposition 4.4. If $A$ is an intuitionistic fuzzy normed prime ideal of $NR$, so the following are satisfied:

i. $\mu_{A}(0_{NR}) = 1$ and $\gamma_{A}(0_{NR}) = 0$;

ii. Im$(\mu_{A}) = \lbrace 1, \alpha \rbrace$ and Im$(\gamma_{A}) = \lbrace 0, \beta \rbrace$, where $\alpha, \beta \in [0, 1]$;

iii. $A_{\ast}$ is a prime ideal of $NR$.

Theorem 4.5. Let $A$ be a fuzzy subset of $NR$ where $A$ is two-valued, $\mu_{A}(0) = 1$ and $\gamma_{A}(0) = 0$, and the set $A_{\ast} = \lbrace r \in NR :\mu_{A}(r) = \mu_{A}(0)$ and $\gamma_{A}(r) = \gamma_{A}(0)\rbrace$ is a prime ideal of $NR$. Hence, $A$ is an intuitionistic fuzzy normed prime ideal of $NR$.

Proof. We have Im$(\mu_{A}) = \lbrace 1, \alpha \rbrace$ and Im$(\gamma_{A}) = \lbrace 0, \beta \rbrace$. Let $r, v \in NR$. If $r, v \in A_{\ast}$, then $r-v \in A_{\ast}$ so, $\mu_{A}(r-v) = 1 = \mu_{A}(r) \ast \mu_{A}(v)$ and $\gamma_{A}(r-v) = 0 = \gamma_{A}(r) \diamond \gamma_{A}(v)$. If $r, v \notin A_{\ast}$, then $\mu_{A}(r-v) = \alpha \geq \mu_{A}(r) \ast \mu_{A}(v)$ and $\gamma_{A}(r-v) = \beta \leq \gamma_{A}(r) \diamond \gamma_{A}(v)$.

Therefore, for all $r, v \in NR$,

Similarly,

Thus $A$ is an intuitionistic fuzzy ideal of $NR$.

Assume $B$ and $C$ be fuzzy ideals of $NR$ where $B \circ C \subseteq A$. Assume that $B \not\subseteq A$ and $C \not\subseteq A$. Then, we have $r, v \in NR$ in such a way that $\mu_{B}(r) > \mu_{A}(r)$ and $\gamma_{B}(r) < \gamma_{A}(r)$, $\mu_{C}(v) > \mu_{A}(v)$ and $\gamma_{C}(r) < \gamma_{A}(r)$, so for all $a \in A_{\ast}$, $\mu_{A}(a) = 1 = \mu_{A}(0)$ and $\gamma_{A}(a) = 0 = \gamma_{A}(0)$, $r \notin A_{\ast}$ and $v \notin A_{\ast}$. Since, $A_{\ast}$ is a prime ideal of $NR$, we have $n \in NR$ in such a way that $rnv \notin A_{\ast}$. Let $a = rnv$ then $\mu_{A}(a) = \mu_{A}(r) = \mu_{A}(v) = \alpha$ and $\gamma_{A}(a) = \gamma_{A}(r) = \gamma_{A}(v) = \beta$, now

and

Which means that $B \circ C \not\subseteq A$. Which contradicts with the hypothesis that $B \circ C \subseteq A$. Therefore, either $B \subseteq A$ or $C \subseteq A$. Then $A$ is an intuitionistic fuzzy normed prime ideal.

Theorem 4.6. Let $P$ be a subset (non-empty) of $NR$. $P$ is a prime ideal if and only if the intuitionistic characteristic function $\lambda_{P} = (\mu_{\lambda_{P}}, \gamma_{\lambda_{P}})$ is an intuitionistic fuzzy normed prime ideal.

Proof. presume that $P$ is a prime ideal of $NR$. So by Theorem 3.12, $\lambda_{P}$ is an intuitionistic fuzzy normed ideal of $NR$. Let $A = (\mu_{A}, \gamma_{A})$ and $B = (\mu_{B}, \gamma_{B})$ be any intuitionistic fuzzy normed ideals of $NR$ with $A \circ B \subseteq \lambda_{P}$ while $A \nsubseteq \lambda_{P}$ and $B \nsubseteq \lambda_{P}$. Then there exist $r, v \in NR$ such that

but

Therefore, $r \notin P$ and $v \notin P$. Since $P$ is a prime ideal, there exist $n \in NR$ such that $rnv \notin P$.

Let $a = rnv$, then $\mu_{\lambda_{P}}(a) = 0$ and $\gamma_{\lambda_{P}}(a) = 1$. Thus, $\mu_{A \circledast b}(a) = 0$ and $\gamma_{A \otimes B}(a) = 1$. but

and

This is a contradiction with $\mu_{\lambda_{P}}(a) = 0$ and $\gamma_{\lambda_{P}}(a) = 1$. Thus for any intuitionistic fuzzy normed ideals $A$ and $B$ of $NR$ we have $A \circ B \subseteq \lambda_{P}$ imply that $A \subseteq \lambda_{P}$ or $B \subseteq \lambda_{P}$. So, $\lambda_{P} = (\mu_{\lambda_{P}}, \gamma_{\lambda_{P}})$ is an intuitionistic fuzzy normed prime ideal of $NR$.

Conversely, suppose $\lambda_{P}$ is an intuitionistic fuzzy normed prime ideal. Let $A$ and $B$ be two intuitionistic fuzzy normed prime ideal of $NR$ such that $A \circ B \subseteq P$. Let $r \in NR$, suppose $\mu_{\lambda_{A} \circledast \lambda_{B}}(r) \neq 0$ and $\gamma_{\lambda_{A} \otimes \lambda_{B}}(r) \neq 1$, then $\mu_{\lambda_{A} \circledast \lambda_{B}}(r) = { \diamond \atop r = cd}(\mu_{\lambda_{A}}(c)\ast \mu_{\lambda_{B}}(d))\neq 0$ and $\gamma_{\lambda_{A} \otimes \lambda_{B}}(r) = { \ast \atop r = cd}(\gamma_{\lambda_{A}}(c)\diamond \gamma_{\lambda_{B}}(d))\neq 1$. Then we have $c, d \in NR$ such that $r = cd$, $\mu_{\lambda_{A}}(c)\neq 0$, $\mu_{\lambda_{B}}(d)\neq 0$ and $\gamma_{\lambda_{A}}(c)\neq 1$, $\gamma_{\lambda_{B}}(d)\neq 1$. Then, $\mu_{\lambda_{A}}(c) = 1$, $\mu_{\lambda_{B}}(d) = 1$ and $\gamma_{\lambda_{A}}(c) = 0$, $\gamma_{\lambda_{B}}(d) = 0$. Which implies $c \in A$ and $d \in B$, therefore $r = cd \in A \circ B \subseteq P$. Then, $\mu_{\lambda_{P}}(r) = 1$ and $\gamma_{\lambda_{P}}(r) = 0$. Thus, for all $r \in NR$, $\mu_{\lambda_{A} \circledast \lambda_{B}}(r) \leq \mu_{\lambda_{P}}(r)$ and $\gamma_{\lambda_{A} \otimes \lambda_{B}}(r)\geq \gamma_{\lambda_{P}}(r)$. So, $\lambda_{A} \circ \lambda_{B} \subseteq \lambda_{P}$. Since $\lambda_{P}$ is an intuitionistic fuzzy normed prime ideal. Then either $\lambda_{A} \subseteq \lambda_{P}$ or $\lambda_{B} \subseteq \lambda_{P}$. Therefore, either $A \subseteq P$ or $B \subseteq P$. Hence $P$ is a prime ideal in $NR$.

Definition 4.7. ^[15] Given a ring $R$ and a proper ideal $M$ of $R$, $M$ is a maximal ideal of $R$ if any of the following equivalent conditions hold:

i. There exists no other proper ideal $J$ of $R$ so that $M \subsetneq J$.

ii. For any ideal $J$ with $M \subseteq J$, either $J = M$ or $J = R$.

Definition 4.8. An intuitionistic fuzzy normed ideal $A$ of a normed ring $NR$ is said to be an intuitionistic fuzzy normed maximal ideal if for any intuitionistic fuzzy normed ideal $B$ of $NR$, $A\subseteq B$, implies that either $B_{*} = A_{*}$ or $B = \lambda_{NR}$. Intuitionistic fuzzy normed maximal left (right) ideal are correspondingly specified.

Proposition 4.9. Let $A$ be an intuitionistic fuzzy normed maximal left (right) ideal of $NR$. Then, $\mid \rm Im\; \mu_{A}\mid$ = $\mid \rm Im\; \gamma_{A}\mid = 2$

Theorem 4.10. Let $A$ be an intuitionistic fuzzy normed maximal left (right) ideal of a normed ring $NR$. Then $A_{\ast} = \lbrace r\in NR :\mu_{A}(r) = \mu_{A}(0)$ and $\gamma_{A}(r) = \gamma_{A}(0)\rbrace$ is a maximal left (right) ideal of $NR$.

Proof. As $A$ is not constant, $A_{\ast} \neq NR$. Then using Proposition 4.9, $A$ is two-valued. Let Im$(\mu_{A}) = \lbrace 1, \alpha \rbrace$ and Im$(\gamma_{A}) = \lbrace 0, \beta \rbrace$, where $0 \leq \alpha < 1$ and $0 < \beta \leq 1$. Assume $M$ to be a left ideal of $NR$ in away that $A_{\ast} \subseteq M$. Take $B$ be an intuitionistic fuzzy subset of $NR$ where if $r \in M$ then $\mu_{B}(r) = 1$ and $\gamma_{B}(r) = 0$ and if $r \notin M$ then $\mu_{B}(r) = c$ and $\gamma_{B}(r) = d$, where $\alpha < c < 1$ and $0 < d < \beta$. Then $B$ is an intuitionistic fuzzy normed left ideal. Obviously $A \subseteq B$. As $A$ is an intuitionistic fuzzy normed maximal left ideal of $NR$ then $A_{*} = B_{*}$ or $B = \lambda_{NR}$. If $A_{*} = B_{*}$ then $A_{*} = M$ given that $B_{*} = M$. If $B = \lambda_{NR}$ subsequently $M = NR$. Therefore, $A_{*}$ is a maximal left ideal of $NR$.

Theorem 4.11. If $A$ is an intuitionistic fuzzy normed maximal left (right) ideal of $NR$, then $\mu_{A}(0) = 1$ and $\gamma_{A}(0) = 0$.

Proof. Suppose $\mu_{A}(0)\neq 1$ and $\gamma_{A}(0)\neq 0$ and $B$ to be an intuitionistic fuzzy subset of $NR$ defined as $B = \lbrace r\in NR:\mu_{B}(r) = h$ and $\gamma_{B}(r) = k\rbrace$, where $\mu_{A}(0) < h < 1$ and $0 < k < \gamma_{A}(0)$. Then, $B$ is an intuitionistic fuzzy normed ideal of $NR$. We can simply check that $A \subset B$, $B \neq \lambda_{NR}$ and $B_{\ast} = \lbrace r\in NR:\mu_{B}(r) = \mu_{B}(0)$ and $\gamma_{B}(r) = \gamma_{B}(0)\rbrace = NR$. Hence, $A \subset B$ but $A_{\ast}\neq B_{\ast}$ and $B \neq \lambda_{NR}$ which contradicts with the assumption that $A$ is an intuitionistic fuzzy normed maximal ideal of $NR$. Therefore, $\mu_{A}(0) = 1$ and $\gamma_{A}(0) = 0$.

Theorem 4.12. Let $A$ be a intuitionistic fuzzy normed left (right) ideal of $NR$. If $A_{\ast}$ is a maximal left (right) ideal of $NR$ with $\mu_{A}(0) = 1$ and $\gamma_{A}(0) = 0$, then $A$ is an intuitionistic fuzzy normed maximal left (right) ideal of $NR$.

Proof. By Proposition 4.9 $A$ is two-valued. Let Im$(\mu_{A}) = \lbrace 1, \alpha \rbrace$ and Im$(\gamma_{A}) = \lbrace 0, \beta \rbrace$, where $0 \leq \alpha < 1$ and $0 < \beta \leq 1$. Define $B$ to be an intuitionistic fuzzy normed left ideal of $NR$ where $A \subseteq B$. Hence, $\mu_{B}(0) = 1$ and $\gamma_{B}(0) = 0$. Let $r \in A_{\ast}$. Then $1 = \mu_{A}(0) = \mu_{A}(r)\leq \mu_{B}(r)$ and $0 = \gamma_{A}(0) = \gamma_{A}(r)\geq \gamma_{B}(r)$. Thus $\mu_{B}(r) = 1 = \mu_{B}(0)$ and $\gamma_{B}(r) = 0 = \gamma_{B}(0)$, hence $r \in B_{\ast}$ then $A_{\ast} \subseteq B_{\ast}$. Given that $A_{\ast}$ a maximal left ideal of $NR$, then $A_{\ast} = B_{\ast}$ or $B_{\ast} = NR$. If $B_{\ast} = NR$ subsequently $B = \lambda_{NR}$. Therefore, $A$ is an intuitionistic fuzzy normed maximal left ideal of $NR$.

Remark 4.13. Let $A \subseteq NR$ and let $0 \leq \alpha \leq 1$ and $0 \leq \beta \leq 1$. Let $\lambda_{A_{\alpha, \beta}}$ be an intuitionistic fuzzy subset of $NR$ where $\mu_{\lambda_{A_{\alpha}}}(r) = 1$ if $r \in A$, $\mu_{\lambda_{A_{\alpha}}}(r) = \alpha$ if $r \notin A$ and $\gamma_{\lambda_{A_{\beta}}}(r) = 0$ if $r \in A$, $\gamma_{\lambda_{A_{\beta}}}(r) = \beta$ if $r \notin A$. If $\alpha = 0$ and $\beta = 1$, the $\lambda_{A_{\alpha, \beta}}$ is the intuitionistic characteristic function of $A$, which identified by $\lambda_{A} = (\mu_{\lambda_{A}}, \gamma_{\lambda_{A}})$. If $NR$ is a ring and $A$ is an intuitionistic fuzzy normed left (right) ideal of $NR$, then:

- $\mu_{\lambda_{A_{\alpha}}}(0) = 1$, $\gamma_{\lambda_{A_{\beta}}}(0) = 0$;

- $({\lambda_{A_{\alpha, \beta}}})_{\ast} = A$, $[({\lambda_{A_{\alpha, \beta}}})_{\ast} = \lbrace r \in NR: \mu_{\lambda_{A_{\alpha}}}(r) = \mu_{\lambda_{A_{\alpha}}}(0)$, $\gamma_{\lambda_{A_{\beta}}}(r) = \gamma_{\lambda_{A_{\beta}}}(0)\rbrace = A]$;

- Im$(\mu_{A}) = \lbrace 1, \alpha \rbrace$ and Im$(\gamma_{A}) = \lbrace 0, \beta \rbrace$;

- $\lambda_{A_{\alpha, \beta}}$ is an intuitionistic fuzzy normed left (right) ideal of $NR$.

5.   Conclusions

In this article, we defined the intrinsic product of two intuitionistic fuzzy normed ideals and proved that this product is a subset of their intersection. Also, we characterized some properties of intuitionistic fuzzy normed ideals. We initiated the concepts of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal and we established several results related to these ideals. Further, we specified the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. We generalised the relation between the intuitionistic characteristic function and prime (maximal) ideals.

Conflict of interest

The author declares no conflict of interest in this paper