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 \(\mathcal {N}\).
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Keywords
- Fuzzy ideals
- Prime fuzzy ideals
- \(\mathcal {G}\)-invariant fuzzy ideals
- \(\mathcal {G}\)-prime fuzzy ideals
2010 Mathematics Subject Classification.
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:
And product \(\eta \circ \mu \) is defined by
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
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
Proof
Take \((h_{1},a_{1})~~and~~ (h_{2},a_{2})\) such that
This implies that \(h_{1}=h_{2}~~ and~~a_{1}=a_{2}.\) Thus, we have
or
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
Also, we have
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
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:
There are only two automorphisms (i) identity map and (ii) the map g defined as follows:
We know that \(Aut(\mathcal {N})\) forms a group. Define a map \(\lambda :\mathcal {N}\rightarrow [0,1]\) by
\(\lambda \) 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.,
This implies that
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
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
i.e.,
and
i.e.,
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
i.e.,
Applying ([5], Lemma 2.3), we obtain
Also,
i.e.,
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
i.e.,
This implies that \(\eta ^{g}\) is a fuzzy left ideal of \(\mathcal {N}.\) Now, for \(r,s ~~and~t\in \mathcal {N},\) we have
i.e.,
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
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
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
Example 3
Let \(\mathcal {X}\) be a near ring. Then
is near ring with regard to matrix addition and matrix multiplication. Let
Then \(\mathcal {I}\) is a fuzzy ideal of \(\mathcal {N}.\) Define a map \(\eta :\mathcal {N}\rightarrow [0,1]\) by
Consider
There are only two automorphisms that are identity map and the map \(g:\mathcal {N}\rightarrow \mathcal {N}\) defined by
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
We prove that \(\eta ^{\mathcal {G}}\) is a fuzzy ideal of \(\mathcal {N.}\)
Let \(r,s \in \mathcal {N}.\) Then
Since \(\eta \) is a fuzzy ideal, we have
This implies that
Also for any \(r,s \in \mathcal {N} \)
Since \(\eta \) is a fuzzy ideal of \(\mathcal {N},\) we have
Thus,
Therefore,
Now,
Again since \(\eta \) is fuzzy ideal, we can write for \(r,s \in \mathcal {N}\)
i.e.,
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 \(\mathcal {N}\).
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 \(g\in \mathcal {G}\)
Also,
This implies that
Thus,
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
is a near ring with regard to matrix addition and matrix multiplication. Let
and
We can check that \(\mathcal {I}_{1}\) and \(\mathcal {I}_{2}\) are ideals of \(\mathcal {N}.\) Define maps
by
and
Then \(\eta _{1}\) and \(\eta _{2}\) are fuzzy ideals of \(\mathcal {N}.\) However
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,
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 \(m,n\in \mathcal {N}\)
Proof
We can easily see that
Now, assume that
And
Then
\(\eta _{r}\) and \(\eta _{s}\) exist in such a way that
or
and
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
This contradicts the fact that
Hence,
Corollary 1
Assume that \(\mathcal {C}=\{\eta _{k}\}\) is a chain of fuzzy ideals of \(\mathcal {N}.\) Then for each \(x_{1},x_{2},...,x_{m}\in \mathcal {N},\)
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
Using Corollary 1, we get
i.e.,
Also,
Again from Corollary 1, we have
i.e.,
Now
Since \(\eta _{k}\) is a fuzzy ideal in \(\mathcal {N}\), we obtain
i.e.,
Again using the fact that \(\eta _{k}\) is fuzzy ideal, we get
Also,
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:
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
\(Aut(Z_{3})\) 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.,
This implies that
and
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
Proof
Let \(\lambda _{1}\circ \lambda _{2}\circ \cdots \circ \lambda _{n}(x)=0.\) Then, there is nothing to demonstrate. Otherwise
Since \(\lambda _{i}\) is a fuzzy ideal of \(\mathcal {N},\) we get
Since \(\mathcal {N}\) is zero symmetric, we have
i.e.,
Also, \(\lambda _{2}\) is a fuzzy ideal; hence,
i.e.,
In a similar manner, we can prove that
Since \(\lambda _{n}\) is a fuzzy ideal in \(\mathcal {N}\), we get
Therefore,
or
or
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
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
Then
or
This implies that
Since \(\mathcal {G}\) is finite, by Corollary 1, we obtain
or
Now by Theorem 1, we get
Thus, we obtain
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
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,
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
Thus, from (29) and (30), we obtain
Let there exist another prime fuzzy ideal \(\sigma \) of \(\mathcal {N}\) such that \(\sigma ^{\mathcal {G}}=\lambda .\) Then
Since \(\mathcal {G}\) is finite, so from Proposition 4, we get
Or for any \(h(\ne g)\in \mathcal {G},\) we have
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
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
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?
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Ali, A., Sharma, R.P., Zishan, A. (2023). Group Action on Fuzzy Ideals of Near Rings. In: Sharma, R.K., Pareschi, L., Atangana, A., Sahoo, B., Kukreja, V.K. (eds) Frontiers in Industrial and Applied Mathematics. FIAM 2021. Springer Proceedings in Mathematics & Statistics, vol 410. Springer, Singapore. https://doi.org/10.1007/978-981-19-7272-0_25
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