Group Action on Fuzzy Ideals of Near Rings

Abstract

In this paper, we introduce the group action on a near ring $\mathcal {N}$ and with it we study group action on fuzzy ideals of $\mathcal {N},$ $\mathcal {G}$-invariant fuzzy ideals, finite products of fuzzy ideals, and $\mathcal {G}$-primeness of fuzzy ideals of $\mathcal {N}$.

1 Introduction

A set $\mathcal {N}$ with two binary operations ‘+’ and ‘$\cdot$’ is known as left near ring if (i) $(\mathcal {N},+)$ is a group (not necessarily abelian), (ii) $(\mathcal {N},\cdot )$ is a semigroup, (iii) $\alpha (\beta +\gamma )=\alpha \cdot \beta +\alpha \cdot \gamma$ $\forall$ $\alpha ,\beta$ and $\gamma$ in $\mathcal {N}.$ Analogously, $\mathcal {N}$ is said to be a right near ring if $\mathcal {N}$ satisfies $(iii)^{'}$ $(\beta +\gamma )\alpha =\beta \cdot \alpha +\gamma \cdot \alpha$ $\forall$ $\alpha ,\beta$ and $\gamma$ in $\mathcal {N}.$ A near ring $\mathcal {N}$ with $0x=0,$ $\forall x\in \mathcal {N},$ is known as zero symmetric if $0x=0,$ (left distributively yields that $x0=0$).Throughout the paper, $\mathcal {N}$ represents a zero symmetric left near ring; for simplicity, we call it a near ring. An ideal of near ring $(\mathcal {N},+,\cdot )$ is a subset $\mathcal {M}$ of $\mathcal {N}$ such that (i) $(\mathcal {M},+)$ $\triangleleft$ $(\mathcal {N},+),$ (ii)$\mathcal{N}\mathcal{M} \subset \mathcal {M},$ (iii) $(n_1+m)n_2-n_{1}n_{2} \in \mathcal {M}$ $\forall$ $m \in \mathcal {M}$ and $n_1,n_2 \in \mathcal {N}.$ Note that if $\mathcal {M}$ fulfils (i) and (ii),  it’s referred to as a left ideal of $\mathcal {N}$. It is termed a right ideal of $\mathcal {N}$ if $\mathcal {M}$ satisfies (i) and (iii). A mapping $\phi :\mathcal {N}\rightarrow \mathcal {N}^{'}$ from near ring $\mathcal {N}$ to near ring $\mathcal {N}^{'}$ is said to be a homomorphism if (i) $\phi (\alpha +\beta )=\phi (\alpha )+\phi (\beta )$ (ii) $\phi (\alpha \beta )=\phi (\alpha )\phi (\beta )$ $\forall$ $\alpha$ and $\beta \in \mathcal {N}$. A homomorphism $\phi :\mathcal {N}\rightarrow \mathcal {N}$ which is bijective is said to be an automorphism on $\mathcal {N}.$ The set of all automorphism of $\mathcal {N}$ denoted by $Aut(\mathcal {N})$ forms a group under the operation of composition of mappings.

The study of group actions on rings led to the establishment of the Galois theory for rings. Lorenz and Passman [12], Montgomery [14], and others researched the skew grouping approach in the context of the Galois theory, as well as the groupring and fixed ring. The link between the $\mathcal {G}$-prime ideals of $\mathcal {R}$ and the prime ideals of skew groupring $\mathcal{R}\mathcal{G}$ was identified by Lorenz and Passman [12]. Montgomery [14] investigated the relationship between the prime ideals of $\mathcal {R}$ and $\mathcal {R^{G}}$, leading him to broaden the scope of the action of a group to spec$\mathcal {R}.$

Fuzzy sets were introduced independently by L.A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set. Liu [11] studied fuzzy ideals of a ring and many researchers [4, 6, 7, 20] extended the concepts.The concept of fuzzy ideals and related features have been applied to a variety of fields, including semigroups, [8,9,10, 18, 19], distributive lattice [2], BCK-algebras [16], and near rings [22]. Kim and Kim [5] defined the exact analogue of fuzzy ideals for near rings.

Sharma and Sharma [19] recently investigated the action of group on the fuzzy ideals of the ring $\mathcal {R}$ and found a relationship between the $\mathcal {G}$-prime fuzzy ideals of $\mathcal {R}$ and the prime fuzzy ideals of $\mathcal {R}$. We define the action of group on a near ring $\mathcal {N}$ and investigate the action of group on fuzzy ideals and $\mathcal {G}$-invariant fuzzy ideals of $\mathcal {N}$, finite products of fuzzy ideals, and $\mathcal {G}$-primeness of fuzzy ideals of $\mathcal {N}$. As a result, we extend Sharma and Sharma’s conclusions to near ring $\mathcal {N}.$

2 Preliminaries

Definition 1

([22]) If $\mathcal {N}$ is a near ring, then a fuzzy set $\tilde{F}$ in $\mathcal {N}$ is a set of ordered pair $\tilde{F}=\{(n,\eta _{\tilde{F}}(n))| n\in \mathcal {N}\},$ $\eta _{\tilde{F}}(n)$ is called membership function.

Definition 2

([22]) Let $\eta$ and $\mu$ be two fuzzy subsets of a near ring $\mathcal {N}.$ Then $\eta \cap \mu$ and $\eta \circ \mu$ are defined as follows:

$$ \eta \cap \mu (m)=min\{\eta (m),\mu (m)\}.$$

And product $\eta \circ \mu$ is defined by

$$\begin{aligned} \eta \circ \mu (m)= & {} {\left\{ \begin{array}{ll} \underset{m=m_{1}m_{2}}{\sup }\left\{ \min (\eta (m_{1}),\mu (m_{2}) ) \right\} &{} \text {if } m=m_{1}m_{2} \\ 0 &{} \text {if } m\ne m_{1}m_{2} . \end{array}\right. } \end{aligned}$$

(1)

Definition 3

([22]) Let $(\mathcal {G},+)$ be a group and $\eta$ be a fuzzy subset of $\mathcal {G}$. Then $\eta$ is fuzzy subgroup if

(i) $\eta (g_{1}+g_{2}) \ge min(\eta (g_{1}),\eta (g_{2})),$ $\forall$ $g_{1},g_{2}$ in $\mathcal {G}$,

(ii) $\eta (g)=\eta (-g),$ $\forall$ g in $\mathcal {G}$.

Definition 4

([22]) A fuzzy subset $\eta$ of a near ring $\mathcal {N}$ is said to be a fuzzy subnear ring of $\mathcal {N}$ if $\eta$ is a fuzzy subgroup of $\mathcal {N}$ with respect to the addition $`+$’ and is a fuzzy groupoid with respect to the multiplication $`\cdot$’, i.e.,

(i) $\eta (x-y) \ge min(\eta (x),\eta (y))$ and (ii) $\eta (xy) \ge min(\eta (x),\eta (y))$ $\forall$ $x,y \in \mathcal {N}.$

Definition 5

([22]) A fuzzy subset $\eta$ of a near ring $\mathcal {N}$ is said to be a fuzzy ideal of $\mathcal {N}$ if $\eta$ satisfies following conditions:

• (i) $\eta$ is fuzzy subnear ring,

• (ii) $\eta$ is normal fuzzy subgroup with respect to $`+$’,

• (iii) $\eta (rs) \ge \eta (s);$ for all r,s in $\mathcal {N},$

• (iv) $\eta ((r+t)s-rs) \ge \eta (t)$; $\forall$ r, s and t in $\mathcal {N}$.

If $\eta$ satisfies (i),(ii), and (iii), then it is called a fuzzy left ideal of $\mathcal {N}$. If $\eta$ satisfies (i),(ii), and (iv), then it is called a fuzzy right ideal of $\mathcal {N}$.

Definition 6

([1]) Let $\mathcal {G}$ be a group and $\mathcal {Z}$ be a set. Then $\mathcal {G}$ is said to act on $\mathcal {Z}$ if there is a mapping $\phi :\mathcal {G}\times \mathcal {Z}\rightarrow \mathcal {Z},$ with $\phi (a,z)$ written $a*z,$ such that

(i) $a*(b*z)=(ab)*z,$    $\forall a,b \in \mathcal {G},$ $z\in \mathcal {Z}$.

(ii) $e*z=z.$$e \in \mathcal {G},$ $z\in \mathcal {Z}$. The mapping $\phi$ is called the action of $\mathcal {G}$ on $\mathcal {Z},$ and $\mathcal {Z}$ is said to be a $\mathcal {G}$-set.

Definition 7

([1]) Let $\mathcal {G}$ be a group acting on a set $\mathcal {Z},$ and let $z\in \mathcal {Z}.$ Then the set

$$\mathcal {G}z=\{az|a\in \mathcal {G}\}$$

is called the orbit of $\mathcal {Z}$ in $\mathcal {G}.$

Proposition 1

Let $\mathcal {N}$ be a near ring and $\mathcal {G}=Aut(\mathcal {N})$, group of all automorphism of $\mathcal {N}$.Then $\mathcal {G}$ acts on $\mathcal {N}$ via following map

$$\phi :\mathcal {G} \times \mathcal {N}\rightarrow \mathcal {N} \text{ which } \text{ is } \text{ defined } \text{ by }~ \phi (h,a)=h(a)~or~say~~h*a=h(a).$$

Proof

Take $(h_{1},a_{1})~~and~~ (h_{2},a_{2})$ such that

$$ (h_{1},a_{1})=(h_{2},a_{2}).$$

This implies that $h_{1}=h_{2}~~ and~~a_{1}=a_{2}.$ Thus, we have

$$h_{1}(a_{1})=h_{2}(a_{1})$$

or

$$\phi (h_{1},a_{1})=\phi (h_{2},a_{2}).$$

Hence, $\phi$ is well defined. Furthermore, we show that $\phi$ is the action of $\mathcal {G}$ on $\mathcal {N}$. Take any $g_{1},g_{2}\in \mathcal {G}~~and~~b\in \mathcal {N}.$ Then

$$\begin{aligned} g_{1}*(g_{2}*b)= & {} g_{1}*(g_{2}(b))=g_{1}(g_{2}(b)) \end{aligned}$$

(2)

$$\begin{aligned} (g_{1}\circ g_{2})*b= & {} (g_{1}\circ g_{2})(b)=g_{1}(g_{2}(b)). \end{aligned}$$

(3)

From (2) and (3), we get

$$\begin{aligned}(g_{1}\circ g_{2})*b&=g_{1}*(g_{2}*b). \end{aligned}$$

Also, we have

$$e*x=x.$$

Hence, $\phi$ is the action of $\mathcal {G}$ on $\mathcal {N}$.

Motivated by the definition of the group action of a finite group on fuzzy ideals of a ring [19], we define a $\mathcal {G}-$fuzzy ideal of $\mathcal {N}$ as follows:

Definition 8

Let $\mathcal {G}$ be a group. Then fuzzy set $\eta$ of $\mathcal {N}$ is a $\mathcal {G}-$set or $\mathcal {G}$ act on $\eta$ if

$$\begin{aligned} \eta ^{g}(r)=\eta (r^{g}),~~~ g\in \mathcal {G} \end{aligned}$$

where $r^{g}$ denotes g acts on r,  $r \in \mathcal {N}.$

Example 1

Let $\mathcal {N}=\{0,1,2\}$ be a set. Then under following two binary operations $\mathcal {N}$ forms a zero symmetric near ring:

$$\begin{aligned} \begin{array}{c|ccc} + &{} 0 &{} 1 &{} 2 \\ \hline 0 &{} 0 &{} 1 &{} 2 \\ 1 &{} 1 &{} 2 &{} 0 \\ 2 &{} 2 &{} 0 &{} 1 \\ \end{array} \qquad \begin{array}{c|ccc} {\cdot } &{} 0 &{} 1 &{} 2 \\ \hline 0 &{} 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 1 &{} 2 \\ 2 &{} 0 &{} 1 &{} 2 \\ \end{array} \end{aligned}$$

$$\begin{aligned} Aut(\mathcal {N})=\{f| f:\mathcal {N}\rightarrow \mathcal {N} ~~is~~isomorphism\}. \end{aligned}$$

There are only two automorphisms (i) identity map and (ii) the map g defined as follows:

$$\begin{aligned} g(0)=0, g(1)=2, and g(2)=1. \end{aligned}$$

We know that $Aut(\mathcal {N})$ forms a group. Define a map $\lambda :\mathcal {N}\rightarrow [0,1]$ by

$$\begin{aligned} \lambda (a)=\Bigg \{\begin{array}{ll} 0.9 &{}~~ a=0\\ 0.8 &{}~~ a=1,2. \\ \end{array} \end{aligned}$$

$\lambda$ is a fuzzy ideal. By Definition 8, $\lambda ^{g}:\mathcal {N}\rightarrow [0,1]$ is defined as $\lambda ^{g}(r)=\lambda (r^{g})$, i.e.,

$$\begin{aligned} \lambda ^{g}(0)&=\lambda (0^{g})=\lambda (0)=0.9 \\ \lambda ^{g}(1)&=\lambda (1^{g})=\lambda (2)=0.8 \\ \lambda ^{g}(2)&=\lambda (2^{g})=\lambda (1)=0.8. \end{aligned}$$

This implies that

$$\begin{aligned} \lambda ^{g}&=\{(0,0.9),(1,0.8),(2,0.8)\}~~~ \text{ and } \end{aligned}$$

(4)

$$\begin{aligned} \lambda ^{e}=\lambda&=\{(0,0.9),(1,0.8),(2,0.8)\}. \end{aligned}$$

(5)

This shows that $\lambda ^{g}$ is a fuzzy ideal of $\mathcal {N},$ since $\lambda =\lambda ^{g}.$

3 Prime Fuzzy Ideals

Definition 9

([19]) Let $\mathcal {Q}$ be a fuzzy ideal of $\mathcal {N}.$ Then $\mathcal {Q}$ is said to be a prime ideal in $\mathcal {N}$ if $\mathcal {Q}$ is not a constant function and for any fuzzy ideals $\eta$ and $\mu$ in $\mathcal {N}$, $\eta \circ \mu \subset \mathcal {Q}$ implies that either $\eta \subset \mathcal {Q}$ or $\mu \subset \mathcal {Q}.$

Example 2

Take $Z_{4}=\{0,1,2,3\}$ the zero symmetric left near ring under binary operations addition modulo 4 and for any $a,b \in Z_{4}$ multiplication is defined as

$$\begin{aligned} a\cdot b=\Bigg \{\begin{array}{ll} b &{} a\ne 0\\ 0 &{} a=0. \\ \end{array} \end{aligned}$$

Define two maps $\eta _{1},\eta _{2}:Z_{4}\rightarrow [0,1]$ by $\eta _{1}(z_{1})=\Bigg \{\begin{array}{ll} 0.9 &{} z_{1}=0\\ 0.8 &{} z_{1}\ne 0, \end{array}$ and $\eta _{2}(z_{2})=0.9$ for all $z_{1},z_{2} \in Z_{4}.$ It shows that $\eta _{1}\circ \eta _{2}\subseteq \eta _{1}$ and $\eta _{1} \subseteq \eta _{1}$ but $\eta _{2} \not \subset \eta _{1}.$ As $\eta _{1}$ is non-constant function so $\eta _{1}$ is a prime fuzzy ideal.

Proposition 2

If $\eta$ is a fuzzy ideal of $\mathcal {N}$, then $\eta ^{g}$ is a fuzzy ideal of $\mathcal {N}$. Moreover, primeness of $\eta$ as a fuzzy ideal implies the primeness of fuzzy ideal $\eta ^{g}$ of $\mathcal {N}$.

Proof

Assume that $\eta$ is a fuzzy ideal of $\mathcal {N}$. Then we show that $\eta ^{g}$ is also a prime fuzzy ideal of $\mathcal {N}$, i.e., we will show that $\eta ^{g}$ satisfies following conditions:

Let $r,s\in \mathcal {N}.$ Since $\eta$ is a fuzzy ideal of $\mathcal {N}$, then we have

$$\begin{aligned} \eta ^{g}(r-s)= & {} \eta (r-s)^{g}=\eta (r^{g}-s^{g})\ge min(\eta (r^{g}),\eta ( s^{g})), \end{aligned}$$

i.e.,

$$\begin{aligned} \eta ^{g}(r-s) \ge min(\eta ^{g}(r),\eta ^{g}(s)) \end{aligned}$$

(6)

and

$$\begin{aligned} \eta ^{g}(rs)&=\eta (rs)^{g}=\eta (r^{g}s^{g})\ge min(\eta (r^{g}),\eta ( s^{g})), \end{aligned}$$

(7)

i.e.,

$$\begin{aligned} \eta ^{g}(rs)\ge min(\eta (r^{g}),\eta ( s^{g})). \end{aligned}$$

(8)

Equations (6) and (7) imply that $\eta ^{g}$ is a fuzzy subnear ring of $\mathcal {N.}$

Again $r,s\in \mathcal {N}$ and $\eta$ is fuzzy ideal of $\mathcal {N},$ we have

$$\begin{aligned} \eta ^{g}(r+s)&=\eta (r+s)^{g}=\eta (r^{g}+s^{g})\ge min(\eta (r^{g}),\eta ( s^{g})), \end{aligned}$$

i.e.,

$$\begin{aligned} \eta ^{g}(r+s)&\ge min(\eta (r^{g}),\eta ( s^{g})). \end{aligned}$$

(9)

Applying ([5], Lemma 2.3), we obtain

$$\begin{aligned} \eta ^{g}(r)&=\eta (r^{g})=\eta (-r^{g})=\eta ^{g}(-r). \end{aligned}$$

Also,

$$\begin{aligned} \eta ^{g}(r)&=\eta (r^{g})=\eta (s^{g}+r^{g}-s^{g})=\eta (s+r-s)^{g}, \end{aligned}$$

i.e.,

$$\begin{aligned} \eta ^{g}(r)&=\eta ^{g}(s+r-s). \end{aligned}$$

(10)

Since $\eta ^{g}$ satisfies all conditions of normal subgroup, $\eta ^{g}$ is a normal fuzzy subgroup of $(\mathcal {N},+).$ For $r,s\in \mathcal {N},$ we have

$$\begin{aligned} \eta ^{g}(rs)&=\eta (rs)^{g}=\eta (r^{g}s^{g})\ge \eta ( s^{g}), \end{aligned}$$

i.e.,

$$\begin{aligned} \eta ^{g}(rs)&\ge \eta ^{g}( s). \end{aligned}$$

(11)

This implies that $\eta ^{g}$ is a fuzzy left ideal of $\mathcal {N}.$ Now, for $r,s ~~and~t\in \mathcal {N},$ we have

$$\begin{aligned} \eta ^{g}((r+t)s-rs)&=\eta ((r^{g}+t^{g})s^{g}-r^{g}s^{g})\ge \eta (t^{g}), \end{aligned}$$

i.e.,

$$\begin{aligned} \eta ^{g}((r+t)s-rs)&\ge \eta ^{g}(t). \end{aligned}$$

(12)

This implies that $\eta$ is a right fuzzy ideal. Thus, $\eta$ is a fuzzy ideal(left fuzzy ideal as well as right fuzzy ideal) of $\mathcal {N}$.

Now we prove that $\eta ^{g}$ is a prime fuzzy ideal of $\mathcal {N}.$ Let $\mathcal {A}$ and $\mathcal {B}$ be two fuzzy ideals of $\mathcal {N}$ such that $\mathcal {A}\circ \mathcal {B}\subset \eta ^{g}$. Then $\mathcal {A}^{g^{-1}}$ and $\mathcal {B}^{g^{-1}}$ are also fuzzy ideals of $\mathcal {N}$, since $g^{-1}\in \mathcal {G}$ and as proved in $\eta ^{g}$, we claim that $\mathcal {A}^{g^{-1}}\circ \mathcal {B}^{g^{-1}}\subset \eta$. Let $n\in \mathcal {N}$ and

$$\begin{aligned} (\mathcal {A}^{g^{-1}}\circ \mathcal {B}^{g^{-1}})(n)= & {} \sup _{n=n_{1}n_{2}}\{ \min (\mathcal {A}^{g^{-1}}(n_{1}),\mathcal {B}^{g^{-1}}(n_{2}) ) \} \\= & {} \sup _{n^{g^{-1}}=n_{1}^{g^{-1}}n_{2}^{g^{-1}}}\left\{ \min (\mathcal {A}(n_{1}^{g^{-1}}),\mathcal {B}(n_{2}^{g^{-1}}) ) \right\} \\= & {} (\mathcal {A}\circ \mathcal {B})(n^{g^{-1}}) \\\le & {} \eta ^{g}(n^{g^{-1}})=\eta ((n^{g^{-1}})^{g})\\= & {} \eta (n). \end{aligned}$$

So, $\mathcal {A}^{g^{-1}}\circ \mathcal {B}^{g^{-1}}\subset \eta$. Since $\eta$ is a prime fuzzy ideal, then we have $\mathcal {A}^{g^{-1}} \subset \eta$ or $\mathcal {B}^{g^{-1}}\subset \eta$. Suppose that $\mathcal {A}^{g^{-1}}\subset \eta .$ Then for all $n\in \mathcal {N},$ we have

$$\begin{aligned} \mathcal {A}(n)&=\mathcal {A}((n^{g})^{g^{-1}})=\mathcal {A}^{g^{-1}}(n^{g})\le \eta (n^{g}) = \eta ^{g}(n). \end{aligned}$$

Thus $\mathcal {A}\subset \eta ^{g}$. This implies that $\eta ^{g}$ is a prime fuzzy ideal of $\mathcal {N}.$

Now we define a $\mathcal {G}$-invariant fuzzy ideal of a near ring.

Definition 10

A fuzzy ideal $\eta$ of $\mathcal {N}$ is called a $\mathcal {G}$-invariant fuzzy ideal of $\mathcal {N}$ if and only if

$$\eta^{g}(r)=\eta (r^{g})\ge \eta (r), \forall g\in \mathcal {G}, r\in \mathcal {N}.$$

$$\text{Or}$$

$$\eta (r)=\eta ((r^{g})^{g^{-1}})\ge \eta (r^{g}).$$

Example 3

Let $\mathcal {X}$ be a near ring. Then

$$N=\Bigg \{\Bigg (\begin{array}{cc} x~ &{}~0 \\ 0~ &{}~y \end{array} \Bigg )\Bigg |~~x,y,~~0\in X\Bigg \}$$

is near ring with regard to matrix addition and matrix multiplication. Let

$$I=\Bigg \{\Bigg (\begin{array}{cc} 0~ &{}~0 \\ 0~ &{}~y \end{array} \Bigg )\Bigg |~~y,0\in X\Bigg \}.$$

Then $\mathcal {I}$ is a fuzzy ideal of $\mathcal {N}.$ Define a map $\eta :\mathcal {N}\rightarrow [0,1]$ by

$$\begin{aligned} \eta (z)=\Bigg \{\begin{array}{ll} 0.9 &{} ~z=0\\ 0.8 &{} ~ z\ne 0 ~~~~~~.\\ \end{array} \end{aligned}$$

Consider

$$\mathcal {G}(\subseteq Aut(\mathcal {N}))=\{f| f:\mathcal {N}\rightarrow \mathcal {N} ~~is~an~isomorphism\}.$$

There are only two automorphisms that are identity map and the map $g:\mathcal {N}\rightarrow \mathcal {N}$ defined by

$$\begin{aligned} g \Bigg (\begin{array}{cc} x~ &{}~0 \\ 0~ &{}~y \end{array} \Bigg ) = \Bigg (\begin{array}{cc} y~ &{}~0 \\ 0~ &{}~x \end{array} \Bigg ). \end{aligned}$$

Since $\eta ^{g}(r)=\eta (r^{g})=\eta (r)~~for~~all~~g\in \mathcal {G}~~ and~~r\in \mathcal {N},$ we get $\eta$ is $\mathcal {G}-$invariant fuzzy ideal in $\mathcal {N}$.

Theorem 1

Let $\eta$ be a fuzzy ideal of $\mathcal {N}$ and $\eta ^{\mathcal {G}}=\bigcap \limits _{g \in \mathcal {G}}\eta ^{g}$. Then $\eta ^{\mathcal {G}}(r)=min\{\eta (r^{g}),g\in \mathcal {G}\}.$ Moreover, fuzzy ideal $\eta$ contains largest $\mathcal {G}$-invariant fuzzy ideal $\eta ^{\mathcal {G}}$ of $\mathcal {N}.$

Proof

Assume that

$$\begin{aligned} \eta ^{\mathcal {G}}(s)= & {} \bigcap \limits _{k \in \mathcal {G}} \eta ^{k} \\ {}= & {} min\{\eta ^{k}(s),~~k\in \mathcal {G}\}=min\{\eta (s^{k}),~~k\in \mathcal {G}\}.\end{aligned}$$

We prove that $\eta ^{\mathcal {G}}$ is a fuzzy ideal of $\mathcal {N.}$

Let $r,s \in \mathcal {N}.$ Then

$$\begin{aligned}\eta ^{\mathcal {G}}(r-s)= & {} min\{ \eta (r-s)^{g},g \in \mathcal {G}\} \\ {}= & {} min\{ \eta (r^{g}-s^{g}),g \in \mathcal {G}\} \\ {}= & {} min\{min (\eta (r^{g}),\eta (s^{g})),g \in \mathcal {G}\}.\end{aligned}$$

Since $\eta$ is a fuzzy ideal, we have

$$\begin{aligned} \eta ^{\mathcal {G}}(r-s)\ge & {} min\{min (\eta (r^{g}),g \in \mathcal {G}),min(\eta (s^{g}),g \in \mathcal {G})\} \\ {}= & {} min\{\eta ^{\mathcal {G}}(r),\eta ^{\mathcal {G}}(s)\}. \end{aligned}$$

This implies that

$$\begin{aligned} \eta ^{\mathcal {G}}(r-s)&\ge \{\eta ^{\mathcal {G}}(r),\eta ^{\mathcal {G}}(s)\}. \end{aligned}$$

(13)

Also for any $r,s \in \mathcal {N}$

$$\begin{aligned}\eta ^{\mathcal {G}}(rs)= & {} min\{ \eta (rs)^{g},g \in \mathcal {G}\} \\ {}= & {} min\{ \eta (r^{g}s^{g}),g \in \mathcal {G}\} \\ {}= & {} min\{min (\eta (r^{g}),\eta (s^{g})),~g \in \mathcal {G}\}.\end{aligned}$$

Since $\eta$ is a fuzzy ideal of $\mathcal {N},$ we have

$$\begin{aligned} \eta ^{\mathcal {G}}(rs)\ge & {} min\{min (\eta (r^{g}),g \in \mathcal {G}),min(\eta (s^{g}),g \in \mathcal {G})\} \\ {}= & {} min\{\mu ^{G}(r),\mu ^{G}(s)\}. \end{aligned}$$

Thus,

$$\begin{aligned} \eta ^{\mathcal {G}}(rs)&\ge \{\eta ^{\mathcal {G}}(r),\eta ^{\mathcal {G}}(s)\}.\end{aligned}$$

(14)

$$\begin{aligned}\eta ^{\mathcal {G}}(s+r-s)= & {} min\{ \eta (s+r-s)^{g},~g \in \mathcal {G}\} \\ {}= & {} min\{ \eta (s^{g}+r^{g}-s^{g}),g \in \mathcal {G}\}\\= & {} min\{\eta (r^{g}),g \in \mathcal {G}\} \\ {}= & {} \eta ^{\mathcal {G}}(r).\end{aligned}$$

Therefore,

$$\begin{aligned} \eta ^{\mathcal {G}}(s+r-s)= & {} \eta ^{\mathcal {G}}(r). \end{aligned}$$

(15)

Now,

$$\begin{aligned}\eta ^{\mathcal {G}}(rs)= & {} min\{ \eta (rs)^{g},g \in \mathcal {G}\} \\ {}= & {} min\{ \eta (r^{g}s^{g}),g \in \mathcal {G}\}. \end{aligned}$$

Again since $\eta$ is fuzzy ideal, we can write for $r,s \in \mathcal {N}$

$$\begin{aligned} \eta ^{\mathcal {G}}(rs)\ge & {} min\{\eta (s^{g}),g \in \mathcal {G}\}. \\ {}= & {} \eta ^{\mathcal {G}}(s), \end{aligned}$$

i.e.,

$$\begin{aligned} \eta ^{\mathcal {G}}(rs)\ge & {} \eta ^{\mathcal {G}}(s) \end{aligned}$$

(16)

$$\begin{aligned} \eta ^{G}((r+t)s-rs)= & {} min\{ \eta ((r+t)s-rs)^{g},g \in \mathcal {G}\} \\ {}= & {} min\{ \eta ((r+t)^{g}s^{g}-r^{g}s^{g}),g \in \mathcal {G}\} \\ {}= & {} min\{ \eta ((r^{g}+t^{g})s^{g}-r^{g}s^{g}),g \in \mathcal {G}\} \\\ge & {} min\{\eta (t^{g}),g \in \mathcal {G}\}. \\ {}= & {} \eta ^{\mathcal {G}}(t) \end{aligned}$$

$$\begin{aligned} \eta ^{\mathcal {G}}((r+t)s-rs)&\ge \eta ^{\mathcal {G}}(t). \end{aligned}$$

(17)

Since $\eta ^{G}$ is the left and right fuzzy ideals of $\mathcal {N}$, then $\eta ^{\mathcal {G}}$ is the fuzzy ideal of $\mathcal {N}$. It is still necessary to show that it is a $\mathcal {G}$-invariant fuzzy ideal of $\mathcal {N}$.

$$\begin{aligned} \eta ^{\mathcal {G}}(r^{g})= & {} \min \{\eta ((r^{g})^{k}), k\in \mathcal {G}\} \\= & {} \min \{\eta (r^{gk}), k\in \mathcal {G}\} \\= & {} \min \{\eta (r^{g^{'}}), g^{'}\in \mathcal {G}\}\\= & {} \eta ^{\mathcal {G}}(r). \end{aligned}$$

Now we prove that $\eta ^{G}$ is the largest. Assume that $\mu$ is any $\mathcal {G}$-invariant fuzzy ideal of $\mathcal {N}$ such that $\mu \subseteq \eta .$ Then for any $g\in \mathcal {G}$

$$\begin{aligned} \mu (r^{g})&=\mu (r) \le \eta (r). \end{aligned}$$

Also,

$$\begin{aligned} \mu (r^{g})&=\mu (r)= \mu ((r^{g})^{g^{-1}})\le \eta (r^{g}). \end{aligned}$$

This implies that

$$\begin{aligned} \mu (r)&\le min\{ \eta (r^{g}), g\in \mathcal {G}\}= \eta ^{\mathcal {G}}(r). \end{aligned}$$

Thus,

$$\begin{aligned} \mu&\subseteq \eta ^{\mathcal {G}}. \end{aligned}$$

Hence, $\eta ^{\mathcal {G}}$ contained in $\eta$ as the largest $\mathcal {G}$-invariant fuzzy ideal of $\mathcal {N}.$

Remark 1

If a fuzzy ideal $\eta$ of $\mathcal {N}$ satisfies $\eta = \eta ^{\mathcal {G}}.$ Then $\eta$ is called as $\mathcal {G}$-invariant fuzzy ideal of $\mathcal {N}$ and vice versa.

4 Union of Fuzzy Ideals of Near Ring

The following example demonstrates that the union of fuzzy ideals of a near ring $\mathcal {N}$ need not be a fuzzy ideal in $\mathcal {N}.$

Example 4

Let $\mathcal {Q}$ be a near ring. Then

$$\begin{aligned} \mathcal {N}=\Bigg \{\Bigg (\begin{array}{cc} 0~ &{}~p \\ 0~ &{}~q \end{array} \Bigg )\Bigg |~~p,q~~0\in \mathcal {Q}\Bigg \} \end{aligned}$$

is a near ring with regard to matrix addition and matrix multiplication. Let

$$\begin{aligned} \mathcal {I}_{1}=\Bigg \{\Bigg (\begin{array}{cc} 0~ &{}~p \\ 0~ &{}~0 \end{array} \Bigg )\Bigg |~~p,~0\in \mathcal {Q}\Bigg \} \end{aligned}$$

and

$$\begin{aligned} \mathcal {I}_{2}=\Bigg \{\Bigg (\begin{array}{cc} 0~ &{}~0 \\ 0~ &{}~q \end{array} \Bigg )\Bigg |~~q,~0\in \mathcal {Q}\Bigg \}. \end{aligned}$$

We can check that $\mathcal {I}_{1}$ and $\mathcal {I}_{2}$ are ideals of $\mathcal {N}.$ Define maps

$$\begin{aligned} \eta _{1}:\mathcal {N}\rightarrow [0,1]~~~~~~ and~~~~~~\eta _{2}:\mathcal {N}\rightarrow [0,1] \end{aligned}$$

by

$$\begin{aligned} \eta _{1}(x)=\Bigg \{\begin{array}{ll} 0.5 &{} x\in \mathcal {I}_{1} \\ 0 &{} x\not \in \mathcal {I}_{1} \\ \end{array} \end{aligned}$$

and

$$\begin{aligned} \eta _{2}(x)=\Bigg \{\begin{array}{ll} 0.6, &{}~ x\in \mathcal {I}_{2} \\ 0, &{} ~ x\not \in \mathcal {I}_{2}. \\ \end{array} \end{aligned}$$

Then $\eta _{1}$ and $\eta _{2}$ are fuzzy ideals of $\mathcal {N}.$ However

$$\begin{aligned} (\eta _{1} \cup \eta _{2}) (x)=\Bigg \{\begin{array}{ll} max\{0.5,0.6\}, &{}~ x\in \mathcal {I}_{1} \cup \mathcal {I}_{2} \\ 0, &{} ~ x\not \in \mathcal {I}_{1} \cup \mathcal {I}_{2} \\ \end{array} \end{aligned}$$

is not a fuzzy ideal of $\mathcal {N}$, since for $m=\Bigg (\begin{array}{cc} 0~ &{}~p \\ 0~ &{}0 \end{array} \Bigg )$ $n=\Bigg (\begin{array}{cc} 0~ &{}~0 \\ 0~ &{}~q \end{array} \Bigg ),$ $m-n=\Bigg (\begin{array}{cc} 0~ &{}~p \\ 0~ &{}-q \end{array} \Bigg ) \notin \mathcal {I}_{1}\cup \mathcal {I}_{2}.$ We see that $\eta _{1} \cup \eta _{2} (m-n)=0,$ $\eta _{1} \cup \eta _{2} (m)=0.6$, and $\eta _{1} \cup \eta _{2} (n)=0.5.$ Thus,

$$\begin{aligned} \eta _{1} \cup \eta _{2} (m-n)=0&\not>\max \{\eta _{1} \cup \eta _{2} (m),\eta _{1} \cup \eta _{2} (n)\} \\&\not>\max \{0.6,0.5\} \\&\not >0.6. \end{aligned}$$

Hence, $\eta _{1} \cup \eta _{2}$ is not a fuzzy ideal of $\mathcal {N}.$

Proposition 3

Let $\mathcal {C}=\{\eta _{k}\}$ be a chain of fuzzy ideals of $\mathcal {N}.$ Then for any $m,n\in \mathcal {N}$

$$\begin{aligned} min(\sup _{k}\{\eta _{k}(m)\},\sup _{k}\{\eta _{k}(n)\})= & {} \sup _{k}\{min (\eta _{k}(m),\eta _{k}(n))\}. \end{aligned}$$

Proof

We can easily see that

$$\begin{aligned} \sup _{k}\{\min (\eta _{k}(m),\eta _{k}(n))\}\le & {} \min (\sup _{k}\{\eta _{k}(m)\},\sup _{k}\{\eta _{k}(n)\}). \end{aligned}$$

Now, assume that

$$\begin{aligned} \sup _{k}\{\min (\eta _{k}(m),\eta _{k}(n))\}= & {} I. \end{aligned}$$

And

$$\begin{aligned} I< & {} \min (\sup _{k}\{\eta _{k}(m)\},\sup _{k}\{\eta _{k}(n)\}). \end{aligned}$$

Then

$$\begin{aligned} \sup _{k}\{\eta _{k}(m)\}>I,~~~~or~~~\sup _{k}\{\eta _{k}(n)\}>I. \end{aligned}$$

$\eta _{r}$ and $\eta _{s}$ exist in such a way that

$$ \begin{aligned} \eta _{r}(m)>I,~~~~ \& ~~~\eta _{s}(n)>I\end{aligned}$$

or

$$\begin{aligned} \eta _{r}(m)>I&\ge \min (\eta _{r}(m),\eta _{r}(n)) \end{aligned}$$

(18)

and

$$\begin{aligned} \eta _{r}(n)>I&\ge \min (\eta _{s}(m),\eta _{s}(n)). \end{aligned}$$

(19)

Since, $\eta _{r},\eta _{s} \in \mathcal {C},$ so without loss of generality, we may assume that $\eta _{r}\subseteq \eta _{s}$ and $\eta _{s}(n)\ge \eta _{s}(m)$ Therefore, from (18) and (19), we get

$$\begin{aligned} I&<\eta _{r}(m)\le \eta _{s}(m)=\min (\eta _{s}(m),\eta _{s}(n)). \end{aligned}$$

This contradicts the fact that

$$\begin{aligned} I= & {} \sup _{k}\{\min (\eta _{k}(m),\eta _{k}(n))\}.\end{aligned}$$

Hence,

$$\begin{aligned} min(\sup _{k}\{\eta _{k}(m)\},\sup _{k}\{\eta _{k}(n)\})= & {} \sup _{k}\{min (\eta _{k}(m),\eta _{k}(n))\}. \end{aligned}$$

Corollary 1

Assume that $\mathcal {C}=\{\eta _{k}\}$ is a chain of fuzzy ideals of $\mathcal {N}.$ Then for each $x_{1},x_{2},...,x_{m}\in \mathcal {N},$

$$\begin{aligned} min(\sup _{k}\{\eta _{k}(x_{1})\},\sup _{k}\{\eta _{k}(x_{2})\},.......,\sup _{k}\{\eta _{k}(x_{m})\})= & {} \sup _{k}\{min (\eta _{k}(x_{1}),\eta _{k}(x_{2}),......,\eta _{k}(x_{m}))\}.\end{aligned}$$

Theorem 1

Let $\mathcal {C}=\{\eta _{k}\}$ be a chain of fuzzy ideals of $\mathcal {N}.$ Then $\bigcup \nolimits _{k} \eta _{k}$ is a fuzzy ideal of $\mathcal {N}.$

Proof

Let $r,s \in \mathcal {N},$ and $\eta _{k}$ be a fuzzy ideal of $\mathcal {N},$ where k is a natural number. Then

$$\begin{aligned} (\bigcup \limits _{k} \eta _{k})(r-s)= & {} \sup _{k}(\eta _{k}(r-s))\\\ge & {} \sup _{k}\{\min (\eta _{k}(r),\eta _{k}(s))\}. \end{aligned}$$

Using Corollary 1, we get

$$\begin{aligned} (\bigcup \limits _{k})(r-s)\ge & {} \min \{\sup _{k}(\eta _{k}(r)),\sup _{k}(\eta _{k}(s))\}, \end{aligned}$$

i.e.,

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(r-s)\ge & {} \min \{(\bigcup \limits _{k}\eta _{k})(r),(\bigcup \limits _{k}\eta _{k})(s)\}. \end{aligned}$$

(20)

Also,

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(rs)= & {} \sup _{k}(\bigcup \limits _{k}(rs))\\\ge & {} \sup _{k}\{\min (\eta _{k}(r),\eta _{r}(s))\}. \end{aligned}$$

Again from Corollary 1, we have

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(rs)\ge & {} \min \{\sup _{k}(\eta _{k}(r)),\sup _{k}(\eta _{k}(s))\}, \end{aligned}$$

i.e.,

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(rs)&\ge \min \{(\bigcup \limits _{k}\eta _{k})(r),(\bigcup \limits _{k}\eta _{k})(s)\}. \end{aligned}$$

(21)

Now

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(s+r-s)= & {} \sup _{k}(\eta _{k}(s+r-s))\\= & {} \sup _{k}\{\eta _{k}(r)\}. \end{aligned}$$

Since $\eta _{k}$ is a fuzzy ideal in $\mathcal {N}$, we obtain

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(s+r-s)= & {} (\bigcup \limits _{k}\eta _{k})(r), \end{aligned}$$

i.e.,

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(s+r-s)= & {} (\bigcup \limits _{k}\eta _{k})(r). \end{aligned}$$

(22)

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(rs)= & {} \sup _{k}(\eta _{k}(rs))\\\ge & {} \sup _{k}\{\eta _{k}(s)\}. \end{aligned}$$

Again using the fact that $\eta _{k}$ is fuzzy ideal, we get

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})(rs)\ge & {} (\bigcup \limits _{k}\eta _{k})(s) \end{aligned}$$

(23)

$$\begin{aligned} (\bigcup \limits _{k}\eta _{k})((r+t)s-rs)= & {} \sup _{k}(\eta _{k}((r+t)s-rs))\\\ge & {} \sup _{k}\{\eta _{k}(t)\}. \end{aligned}$$

Also,

$$\begin{aligned} ( \bigcup \limits _{k}\eta _{k})((r+t)s-rs)\ge & {} ( \bigcup \limits _{k}\eta _{k})(t). \end{aligned}$$

(24)

Hence, $(\bigcup \limits _{k}\eta _{k})$ is a fuzzy ideal of $\mathcal {N}.$

5 G-Prime Fuzzy Ideals of a Near Ring

Motivated by the definition of $\mathcal {G}$-prime fuzzy ideals of the rings [19], we define $\mathcal {G}-$prime fuzzy ideals in a near ring as follows.

Definition 11

Let the fuzzy ideal $\eta$ of $\mathcal {N}$ be $\mathcal {G}$-invariant and non-constant. If $\mu \circ \lambda \subseteq \eta$ implies that either $\mu \subseteq \eta$ or $\lambda \subseteq \eta$ for any two $\mathcal {G}$-invariant fuzzy ideals $\mu$ and $\lambda$ of $\mathcal {N}$, then $\eta$ is a $\mathcal {G}$-prime fuzzy ideal.

Example 5

Take $Z_{3}=\{0,1,2\}$ which is a zero symmetric left near ring under binary operations addition modulo 3 and for any $r,s \in Z_{3}$ multiplication is defined as follows:

$$\begin{aligned} r\cdot s=\Bigg \{\begin{array}{ll} s &{} ~~r\ne 0\\ 0 &{} ~~ r=0. \\ \end{array} \end{aligned}$$

$$Aut(Z_{3})=\{f| f:Z_{3}\rightarrow Z_{3} ~~is~~isomorphism\}.$$

We can check that there are only two automorphisms on $Z_3$; one is the identity map and the other is the map g defined by

$$g(0)=0, g(1)=2 \,\,\text{and}\,\, g(2)=1.$$

$Aut(Z_{3})$ forms a group under the composition of mappings. Now we define two maps $\eta _{1},\eta _{2}:Z_{3}\rightarrow [0,1]$ by $\eta _{1}(r)=\Bigg \{\begin{array}{ll} 0.9 &{} ~~r=0\\ 0.8 &{}~~ r\ne 0, \end{array}$ and $\eta _{2}(s)=0.9$ for all $r,s \in Z_{3}.$ By Definition 8, $\eta _{1}^{g}:Z_{3}\rightarrow [0,1]$ is defined as $\eta _{1}^{g}(r)=\eta _{1}(r^{g})$, i.e.,

$$\begin{aligned} \eta _{1}^{g}(0)&=\eta _{1}(0^{g})=\eta _{1}(0)=0.9 \\ \eta _{1}^{g}(1)&=\eta _{1}(1^{g})=\eta _{1}(2)=0.8 \\ \eta _{1}^{g}(2)&=\eta _{1}(2^{g})=\eta _{1}(1)=0.8. \end{aligned}$$

This implies that

$$\begin{aligned} \eta _{1}^{g}&=\{(0,0.9),(1,0.8),(2,0.8)\} \end{aligned}$$

(25)

and

$$\begin{aligned} \eta _{1}^{e}=\eta _{1}&=\{(0,0.9),(1,0.8),(2,0.8)\}. \end{aligned}$$

(26)

Also, we can see that $\eta _{2}$ is a $\mathcal {G}$-invariant fuzzy ideal of $Z_{3}.$ Since $\eta _{1}\circ \eta _{2}\subseteq \eta _{1}$ and $\eta _{1} \subseteq \eta _{1}$ but $\eta _{2} \not \subset \eta _{1},$ so it follows that $\eta _{1}$ is $\mathcal {G}$-prime fuzzy ideal as $\eta _{1}$ is non-constant function.

The following proposition is an extension of Lemma 2.6 of [22] in case of near rings:

Proposition 4

If $\mathcal {N}$ is near ring and $\lambda _{1},\lambda _{2},...,\lambda _{n}$ are fuzzy ideals of $\mathcal {N},$ then

$$\begin{aligned}\lambda _{1}\circ \lambda _{2}\circ \cdots \circ \lambda _{n}&\subset \lambda _{1}\bigcap \lambda _{2}\bigcap \cdots \bigcap \lambda _{n}. \end{aligned}$$

Proof

Let $\lambda _{1}\circ \lambda _{2}\circ \cdots \circ \lambda _{n}(x)=0.$ Then, there is nothing to demonstrate. Otherwise

$$\begin{aligned}\lambda _{1}\circ \lambda _{2}\circ \cdots \circ \lambda _{n}(x)=\sup _{x=x_{1}x_{2}\cdots x_{n}} \{ \min (\lambda _{1}(x_{1}),\lambda _{2}(x_{2}),....,\lambda _{n}(x_{n}))\}. \end{aligned}$$

Since $\lambda _{i}$ is a fuzzy ideal of $\mathcal {N},$ we get

$$\begin{aligned} \lambda _{i}((x+z)y-xy)&\ge \lambda _{i}(z). \end{aligned}$$

Since $\mathcal {N}$ is zero symmetric, we have

$$\begin{aligned} \lambda _{1}(x)= \lambda _{1}(x_{1}x_{2}\cdots x_{n})= & {} \lambda _{1}((0+x_{1})x_{2}\cdots x_{n}-0\cdot x_{1}x_{2}\cdots x_{n}).\\\ge & {} \lambda _{1}(x_{1}), \end{aligned}$$

i.e.,

$$\begin{aligned} \lambda _{1}(x)&\ge \lambda _{1}(x_{1}). \end{aligned}$$

Also, $\lambda _{2}$ is a fuzzy ideal; hence,

$$\begin{aligned} \lambda _{2}(x)= \lambda _{2}(x_{1}x_{2}\cdots x_{n})\ge \lambda _{2}(x_{2}x_{3}\cdots x_{n})= & {} \lambda _{2}((0+x_{2})x_{3}\cdots x_{n}-0\cdot x_{2}x_{3}\cdots x_{n}).\\\ge & {} \lambda _{2}(x_{2}), \end{aligned}$$

i.e.,

$$\begin{aligned} \lambda _{2}(x)&\ge \lambda _{2}(x_{2}). \end{aligned}$$

In a similar manner, we can prove that

$$\begin{aligned} \lambda _{3}(x)&\ge \lambda _{3}(x_{3}), \end{aligned}$$

$$\begin{aligned} \lambda _{4}(x)&\ge \lambda _{4}(x_{4}), \end{aligned}$$

$$\cdots $$

$$\cdots $$

$$\cdots $$

$$\begin{aligned} \lambda _{n-1}(x)&\ge \lambda _{n-1}(x_{n-1}). \end{aligned}$$

Since $\lambda _{n}$ is a fuzzy ideal in $\mathcal {N}$, we get

$$\begin{aligned} \lambda _{n}(x)&\ge \lambda _{n}(x_{n}). \end{aligned}$$

Therefore,

$$\begin{aligned} \lambda _{1}\circ \lambda _{2}\circ \cdots \circ \lambda _{n}(x)&= \min (\lambda _{1}(x_{1}),\lambda _{2}(x_{2}),....,\lambda _{n}(x_{n})) \end{aligned}$$

or

$$\begin{aligned} \underset{1\le i\le n}{\circ }\lambda _{i}(x)\le & {} (\bigcap \limits _{1\le i\le n} \lambda _{i})(x) \end{aligned}$$

or

$$\begin{aligned} \underset{1\le i\le n}{\circ }\ \lambda _{i} \subset \bigcap \limits _{1\le i\le n} \lambda _{i}. \end{aligned}$$

Now we will prove the main result.

Theorem 2

If $\eta$ is a prime fuzzy ideal of $\mathcal {N}.$ Then $\eta ^{G}$ is a $\mathcal {G}$-prime fuzzy ideal of $\mathcal {N}.$ Conversely, if $\lambda$ is a $\mathcal {G}$-prime fuzzy ideal of $\mathcal {N},$ then there exists a prime fuzzy ideal $\eta$ of $\mathcal {N}$ such that $\eta ^{\mathcal {G}}=\lambda ,$ $\eta$ is unique up to its $\mathcal {G}$-orbit.

Proof

Assume that $\eta$ is a prime fuzzy ideal of $\mathcal {N}$ and $\mathcal {P},$ $\mathcal {Q}$ are two $\mathcal {G}$-invariant fuzzy ideals of $\mathcal {N}$ such that $\mathcal {P\circ Q}\subseteq \eta ^{\mathcal {G}}.$ Since $\eta ^{\mathcal {G}}$ is the largest $\mathcal {G}$-invariant fuzzy ideal contained in $\eta ,$ then $\mathcal {P\circ Q}\subseteq \eta .$ Also primeness of $\eta$ implies that either $\mathcal {P}\subseteq \eta$ or $\mathcal {Q}\subseteq \eta .$ Therefore, by Theorem 1 either $\mathcal {P}\subseteq \eta ^{\mathcal {G}}$ or $\mathcal {Q}\subseteq \eta ^{\mathcal {G}}.$ Thus, $\eta ^{\mathcal {G}}$ is a $\mathcal {G}$-prime fuzzy ideal.

Conversely, suppose that $\lambda$ is a $\mathcal {G}$-prime fuzzy ideal of $\mathcal {N}$ and consider

$$\mathcal {S}=\{\eta ,\text{ a } \text{ fuzzy } \text{ ideal } \text{ of } \text{ N }|~~ \eta ^{\mathcal {G}}\subseteq \lambda \}.$$

Before using Zorn’s lemma on $\mathcal {S}$ to get the maximal element(i.e., maximal ideal), we have to show that if $\mathcal {C}=\{\eta _{k}\}\subset \mathcal {S}$ is a chain in $\mathcal {S},$ then $\bigcup \limits _{k}\eta _{k} \in \mathcal {S}.$

Now, from Theorem 1, $\bigcup _{k}\eta _{k}$ is a fuzzy ideal of $\mathcal {N}.$ Since $\eta _{k}\in S,$ we get $\eta ^{\mathcal {G}}_{k} \subseteq \lambda ,$ and we can take any $r\in \mathcal {N}$ and $\eta _{k} \in \mathcal {C}$ such that

$$\begin{aligned} \eta ^{g}_{k}(r)=\eta _{k}(r^{g})~~~~ \text{ and }~~~~ \eta ^{g}_{k} \subseteq \lambda . \end{aligned}$$

Then

$$\begin{aligned} \eta _{k}(r^{g})&=\eta _{k}^{g}(r) \le \lambda (r), \end{aligned}$$

or

$$\begin{aligned} \min (\eta _{k}(r^{g}), g\in \mathcal {G})&\le \lambda (r). \end{aligned}$$

This implies that

$$\begin{aligned} \sup \{\min (\eta _{k}(r^{g}), g\in \mathcal {G})\}&\le \lambda (r). \end{aligned}$$

(27)

Since $\mathcal {G}$ is finite, by Corollary 1, we obtain

$$\begin{aligned} \min \{\sup (\eta _{k}(r^{g}), g\in \mathcal {G})\}= & {} \sup _{k}\{\min (\eta _{k}(r^{g}), g\in \mathcal {G})\}. \end{aligned}$$

(28)

From (27) and (28), we have

$$\begin{aligned} \min \{{\sup _{k}}(\eta _{k}(r^{g}), g\in \mathcal {G})\}&\le \lambda (r) \end{aligned}$$

or

$$\begin{aligned} \min \{({\bigcup \limits _{k}}\eta _{k})(r^{g}), g\in \mathcal {G}\}&\le \lambda (r). \end{aligned}$$

Now by Theorem 1, we get

$$\begin{aligned} ({\bigcup \limits _{k}}\eta _{k})^{\mathcal {G}}(r)&\le \lambda (r). \end{aligned}$$

Thus, we obtain

$$\begin{aligned} ({\bigcup \limits _{k}}\eta _{k})^{\mathcal {G}}&\subseteq \lambda . \end{aligned}$$

This shows that $({\bigcup \limits _{k}}\eta _{l}) \in \mathcal {S}$, i.e., $\mathcal {S}$ has upper bound. Now we use Zorn’s lemma on $\mathcal {S}$ to choose a maximal fuzzy ideal say $\eta .$ Let $\mathcal {P},$ $\mathcal {Q}$ be fuzzy ideals of $\mathcal {N}$ such that $\mathcal {P\circ Q} \subseteq \eta .$ Then

$$\begin{aligned} (\mathcal {P}\circ \mathcal {Q})^{\mathcal {G}}\subseteq \eta ^{\mathcal {G}}\subseteq \lambda . \end{aligned}$$

(29)

Since $\mathcal {P}^{\mathcal {G}}$ and $\mathcal {Q}^{\mathcal {G}}$ are the largest fuzzy ideals contained in $\mathcal {P}$ and $\mathcal {Q}$, respectively.

Now we prove that $\mathcal {P}^{\mathcal {G}}\circ \mathcal {Q}^{\mathcal {G}}\subseteq \mathcal {P\circ Q}$ is a $\mathcal {G}$-invariant,

$$\begin{aligned} (\mathcal {P}^{\mathcal {G}}\circ \mathcal {Q}^{\mathcal {G}})(r^{g})= & {} {\sup _{r^{g}=ab}} \{\min (\mathcal {P}^{\mathcal {G}}(a),\mathcal {Q}^{\mathcal {G}}(b))\} \\ {}= & {} {\sup _{r=a^{g^{-1}}b^{g^{-1}}}} \{\min (\mathcal {P}^{\mathcal {G}}(a^{g^{-1}}),\mathcal {Q}^{\mathcal {G}}(b^{g^{-1}}))\} \\= & {} \mathcal {P}^{\mathcal {G}}\circ \mathcal {Q}^{\mathcal {G}}(r). \end{aligned}$$

Hence, by Theorem 1, $(\mathcal {P}^{\mathcal {G}}\circ \mathcal {Q}^{\mathcal {G}})\subseteq (\mathcal {P\circ Q})^{\mathcal {G}}\subseteq \lambda$. Since $\lambda$ is $\mathcal {G}$-prime, then we have either $\mathcal {P}^{\mathcal {G}}\subseteq \lambda$ or $\mathcal {Q}^{\mathcal {G}}\subseteq \lambda .$ By maximality of $\eta$ either $\mathcal {P}\subseteq \eta$ or $\mathcal {Q}\subseteq \eta .$ This implies that $\eta$ is prime fuzzy ideal of $\mathcal {N}.$ As $\lambda ^{\mathcal {G}}=\lambda ,$ we have $\lambda \in \mathcal {S}.$ But maximality of $\eta$ gives that $\lambda \subseteq \eta .$ Since $\lambda$ and $\eta ^{\mathcal {G}}$ are $\mathcal {G}$-invariant ideal and $\eta ^{\mathcal {G}}$ is largest in $\eta ,$ we get

$$\begin{aligned} \lambda \subseteq \eta ^{\mathcal {G}}. \end{aligned}$$

(30)

Thus, from (29) and (30), we obtain

$$\begin{aligned} \eta ^{\mathcal {G}}=\lambda . \end{aligned}$$

Let there exist another prime fuzzy ideal $\sigma$ of $\mathcal {N}$ such that $\sigma ^{\mathcal {G}}=\lambda .$ Then

$$\begin{aligned} {\bigcap \limits _{g\in \mathcal {G}}}\eta ^{g}=\eta ^{\mathcal {G}}=\sigma ^{G}\subseteq \sigma . \end{aligned}$$

Since $\mathcal {G}$ is finite, so from Proposition 4, we get

$$\begin{aligned} \underset{g\in \mathcal {G}}{\circ }\eta ^{g}\subseteq {\bigcap \limits _{g\in \mathcal {G}}}\eta ^{g}. \end{aligned}$$

Or for any $h(\ne g)\in \mathcal {G},$ we have

$$\begin{aligned} \eta ^{h}\circ ({\bigcap \limits _{\begin{array}{c} g\in \mathcal {G}\\ g \ne h \end{array}}}\eta ^{g})\subseteq {\bigcap \limits _{g\in \mathcal {G}}}\eta ^{g} \subseteq \sigma . \end{aligned}$$

By fuzzy primeness either $\eta ^{h}\subseteq \sigma$ or $\bigcap \nolimits _{\begin{array}{c} g\in \mathcal {G}\\ g \ne h \end{array}}\eta ^{g} \subseteq \sigma .$ If $\eta ^{h}\subseteq \sigma ,$ then $\eta \subseteq \sigma ^{h^{-1}}$ and maximality of $\eta$ with $(\sigma ^{h^{-1}})^{\mathcal {G}}\subseteq \lambda$ implies that

$$\begin{aligned} \eta =\sigma ^{h^{-1}}. \end{aligned}$$

(31)

On the other hand, if $\eta ^{h} \not \subseteq \sigma ,$ we get $\bigcap \nolimits _{\begin{array}{c} g\in \mathcal {G}\\ g \ne h \end{array}}\eta ^{g}\subseteq \sigma .$ Thus, there exists some $(h\ne )g \in \mathcal {G}$ such that $\eta ^{g}\subseteq \sigma$ and hence $\eta \subseteq \sigma ^{g^{-1}}.$ Again maximality of $\eta$ with $(\sigma ^{g^{-1}})^{\mathcal {G}}\subseteq \lambda$ yields that

$$\begin{aligned} \eta =\sigma ^{g^{-1}}. \end{aligned}$$

(32)

Equations (31) and (32) show that $\eta$ is unique up to its $\mathcal {G}$-orbit.

Conclusion: In the future, we plan to study partial group action (the existence of $g*(h*x)$ implies the existence of $(gh)*x,$ but not necessarily conversely) on fuzzy ideals of near rings. The theorems that we prove are the following which are generalizations of Theorems 1 and 2.

Open Problem 1. Can we establish relation between $\mathcal {G}-$invariant fuzzy ideal and largest $\mathcal {G}-$invariant fuzzy ideal of $\mathcal {N}$ under partial group action?

Open Problem 2. Can we investigate relationship between primeness and $\mathcal {G}$-primeness of fuzzy ideal if a group $\mathcal {G}$ partially acts on a fuzzy ideal?