Received 27 December 2015; accepted 15 March 2016; published 18 March 2016

1. Introduction and Main Results
The concept of fuzzy subset of a non-empty set was introduced by Zadeh [1] who introduced the notion of a fuzzy set as a method of representing uncertainty in real physical world. Following this landmark discovery, a number of studies of Fuzzy Modules and their applications have emerged. In particular, Negoita and Ralescu in [2] introduced and examined the notion of a fuzzy submodule of a module. Since then, different types of fuzzy submodules have been investigated in the last three decades. Juncheol Han in [3] has studied group actions in regular rings. The notion of group action on fuzzy subset of a ring was defined and studied by Sharma in [4] [5] .
Let X be a non-empty set. A mapping µ: X ® [0, 1] is called a fuzzy subset of X. Rosenfeld [6] applied the concept of fuzzy sets to the theory of groups and defined the concept of fuzzy subgroups of a group. Since then, many papers concerning various fuzzy algebraic structures have appeared in the literature. As a generalization of a fuzzy set, the concept of an intuitionistic fuzzy set was introduced by Atanassov [7] . Further results on these and other aspects of fuzzy modules can be found in [8] - [17] .
In this paper, we define the group action on fuzzy subset of a module over the ring of integers and introduce the notion of fuzzy G-modules. Many properties of fuzzy G-modules will be established. The concept of fuzzy G-prime submodules will be introduced and studied. Following the definition of G-invariant submodule of a module M, we define and study G-invariant fuzzy submodule and G-invariant fuzzy prime submodule of a module M. The homomorphic image and pre-image of fuzzy G-modules will be established. A number of associated results will be obtained.
2. Preliminaries Knowledge and Results
We recall some definitions and results for the smooth flow of our assertions and results. Throughout the paper, unless otherwise stated, R will denote a commutative ring with unity and M an R-module.
Definition (2.1) [18] Let
and
be any two fuzzy sets of an R-module, then
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Definition (2.2) [19] Let M be an R-module. Then the fuzzy set
of M is called a fuzzy submodule (FSM) of M if
(i) 
(ii) 
(iii) 
Definition (2.3) [19] Let m and n be two fuzzy submodules of an R-module M, then their sum m + n and product mn are defined as
(i) 
(ii) 
Theorem (2.4) [19] Let m and n be two fuzzy submodules of an R-module M. Then the sum m + n and the product mn of m and n are also fuzzy submodules of M.
Theorem (2.5) [20] Let m and n be two fuzzy submodules of an R-module M. Then m Ç n is also a fuzzy submodule of M.
In particular, if {mi: iÎI} be a family of fuzzy submodules of an R-module M, then
is also a fuzzy submodule of M.
Theorem (2.6) [20] Let m andnbe two fuzzy submodules of an R-module M. Then m × n is also a fuzzy submodule of M ´ M.
3. Group Action on Fuzzy Modules
Most of the results below can be extended to an arbitrary commutative ring. We have not been able to remove the restriction to the ring of integers in Lemma (3.3) (iii) and so some results cannot be extended.
Let M be a module over the ring of integers Z and G a finite group which acts on M (i.e., "g Î G, x Î M, xg = gxg−1 Î M). The identity element of G is denoted by e.
Definition (3.1) A group action of G on a fuzzy set 𝜇of a Z-module M is denoted by mg and is defined by
From the definition of group action G on a fuzzy set, following results are easy to verify.
Lemma (3.2) Let m and n be two fuzzy sets of Z-module M and G a finite group which acts on M. Then
(i) ![]()
(ii) ![]()
(iii) ![]()
(iv) ![]()
(v) ![]()
(vi) ![]()
Let us now prove
Lemma (3.3) Let G be a finite group which acts on Z-module M. Then for every x, yÎ M, gÎ G and rÎ Z, we
(i)
(ii)
(iii)
(iv) ![]()
Proof: (i) ![]()
(ii) ![]()
(iii) ![]()
(iv) ![]()
Proposition (3.4) Let m be a fuzzy submodule of Z-module M and G a finite group which acts on M, then mg is also a fuzzy submodule of M.
Proof: Clearly, ![]()
Let
and
then, by lemma (3.3) (i),
![]()
Moreover,
by lemma (3.3) (iii).
Hence
is a fuzzy submodule of M.
Remark (3.5) The converse of Proposition (3.4) does not hold.
Example (3.6) Let M = (Z4 = {0, 1, 2, 3}, +4, ´4) regarded as Z-module, and a finite group G = ({1, 2, 3, 4}, ´5). Consider a fuzzy set 𝜇 of M given by μ(0) = 1, μ(1) = 0.4, μ(2) = 0.6, μ(3) = 0.5. Clearly μ is not fuzzy submodule of M because
.
Take g = 2 so that g−1 = 3, then
, we get ![]()
Now, it is easy to check that μg is a fuzzy submodule of M.
Definition (3.7) Let m be a fuzzy set of Z-module M and G be a finite group which acts on M. Then m is called a fuzzy G-submodule of M if mg is fuzzy submodule of M for all gÎG.
Remark (3.8) (i) From definition (3.7) we see that every fuzzy G-submodule is also a fuzzy submodule, for
.
(ii) Note that the fuzzy set μ in example (3.6) is not a fuzzy G-submodule of M, for
is not a fuzzy submodule of M.
Theorem (3.9) Let m be a fuzzy submodule of Z-module M and G be a finite group which acts on M, then m is a fuzzy G-submodule of M if and only if for every gÎG,
satisfies the following conditions:
(i)
; (ii) ![]()
Proof: Firstly, let m be a FSM of Z-module M and g Î G such that mg satisfies the given conditions.
Substituting r = s = 1 in condition (ii), we get
![]()
Also, putting s = 0 in condition (ii) we get
![]()
Thus ![]()
Therefore,
is a fuzzy submodule of M and hence
is a fuzzy G-submodule of M.
Conversely, let m be a fuzzy G-submodule of M, and gÎG. To establish (i) and (ii), we only need to prove (ii).
Let r, s Î Z and x, y Î M. Then
![]()
Thus
Hence the result proved.
Proposition (3.10) Let m and n be two fuzzy G-submodules of a Z-module M and G be a finite group which acts on M. Then mÇn is also a fuzzy G-submodule of M.
Proof: Since m and n are fuzzy G-submodules of M. Therefore
and
are fuzzy submodules of M for all g Î G which implies that
is fuzzy submodule of M [By Theorem (2.5)].
Thus
is fuzzy submodule of M for all g Î G [By Lemma (3.2) (ii)]. Hence m Ç n is also a fuzzy G-submodule of M.
Proposition (3.11) Let m and n be two fuzzy G-submodules of Z-module M and G be a finite group which acts on M. Then m ×n is also a fuzzy G-submodule of M ´ M.
Proof: Since m and n are fuzzy G-submodules of M. Therefore
and
are fuzzy submodules of M for all g Î G which implies that
is fuzzy submodule of M ´ M [By Theorem (2.6)]. Thus
is a
fuzzy submodule of M ´ M for all gÎG [By Lemma (3.2) (iv)]. Hence m × n is also a fuzzy G-submodule of M ´ M.
We now define fuzzy prime submodule (FPSM) and fuzzy G-prime submodule of M.
Definition (3.12) A fuzzy submodule g of Z-module M is called a fuzzy prime submodule if for any two fuzzy submodules m,n of M such that mn Í g implies that either m Í g or n Í g.
Lemma (3.13) Let m and n be two fuzzy prime submodules of Z-module M. Then mÇn is fuzzy prime submodule of M if and only if either m Í n or n Í m.
Proof. We know that mn Í m Ç n. Therefore, m Ç n is fuzzy prime submodule of M if and only if either m Í m Ç n or n Í m Ç n. But mÇn Í m and m Ç n Í m. Thus m Ç n is fuzzy prime submodule of M if and only if either m = m Ç n or n = m Ç n, i.e., either m Í n or n Í m.
Remark (3.14) From Lemma (3.13) we infer that in general intersection of two fuzzy prime submodules need not to be a fuzzy prime submodule.
Theorem (3.15) Let g be a fuzzy prime submodule of a Z-module M and G be a finite group which acts on M. Then gg is also a fuzzy prime submodule of M.
Proof: Let m andn be fuzzy submodules of M such that mn Í gg. Now, we claim that
. It is sufficient to show that ![]()
Now
![]()
Thus
which implies
. But
is a fuzzy prime submodule of M. Therefore, either
or
which amounts to
or
. Hence
is a fuzzy prime submodule of M.
Definition (3.16) Let g be a fuzzy prime submodule of Z-module M and G be a finite group which acts on M. Then g is called a fuzzy G-prime submodule of M if gg is fuzzy prime submodule of M for all gÎ G.
Remark (3.17) If we denote
, where g is a fuzzy prime submodule of M. Then gG need not be a
fuzzy prime submodule of M, because intersection of fuzzy prime submodules of M, in general, is not a fuzzy prime submodule of M (See Remark (3.14)).
Definition (3.18) A fuzzy submodule g of Z-module M is called a fuzzy semi prime submodule (FSPSM) if for all fuzzy submodules m of M such that m2 Í g implies that m Í g.
Theorem (3.19) Intersection of two fuzzy prime submodules of a Z-module M is always a fuzzy semi prime submodule of M.
Proof. Let g1, g2 be two fuzzy prime submodule of a Z-module M. Let m be a fuzzy submodule of M such that m2 Í g1 Ç g2. Then we have m2 Í g1 and m2 Í g2. But g1 and g2 are FPSMs of M. Therefore, m Í g1 and m Í g2 which implies that m Í g1 Ç g2. Hence g1 Ç g2 is fuzzy semi prime submodule of M.
Theorem (3.20) Let g be a fuzzy semi prime submodule of a Z-module M and G be a finite group which acts on M. Then gg is also a fuzzy semi prime submodule of M.
Proof. Let m be any fuzzy submodule of M and g Î G be any element such that m2 Í gg, then
[follows from Theorem (3.15)], but
is fuzzy semi prime submodule. Therefore,
, implying
. Hence
is fuzzy semi prime submodule of M.
Definition (3.21) Let g be a fuzzy semi prime submodule of Z-module M and G be a finite group which acts on M. Then g is called a fuzzy G-semi prime submodule (FGSPSM) of M ifgg is fuzzy semi prime submodule of M for all g Î G.
Theorem (3.22) If we denote
, where g is a fuzzy semi prime submodule of M. Then gG is a fuzzy G-semi prime submodule of M.
Proof. Let m be a fuzzy submodule of M such that
, "g Î G Þµ Í gg ,"g Î G
. Hence gG is a fuzzy G-semi prime submodule of M.
Following the definition of G-invariant submodule of a module M, we define G-invariant fuzzy submodule and G-invariant fuzzy prime submodule of Z-module M.
Definition (3.23) Let m be a fuzzy submodule of a Z-module M and G be a finite group. Then m is said to be G-invariant fuzzy submodule of M if and only if ![]()
Proposition (3.24) Let m be a fuzzy submodule of a Z-module M and G be a finite group which acts on M. Then m is G-invariant fuzzy submodule of M if and only if
, "g Î G.
Proof: For xÎM, gÎG, we have
implies that "g Î G, we have
. Hence
.
Lemma (3.25) Let m be a fuzzy submodule of a Z-module M and G be a finite group which acts on M. Let
where
. Then mG is the largest G-invariant fuzzy submodule of M.
Proof: Since m be a fuzzy submodule of Z-module of M and so mg is fuzzy submodule of M for all g Î G. Also,intersection of fuzzy submodules of M is a fuzzy submodule of M. Now,
![]()
Therefore,
Thus mG is G-invariant fuzzy submodule of M. Further, let n be any G-invariant fuzzy submodule of M such thatn Í m. Then for gÎG, we get
Now,
Thus
Hence
is the largest G-invariant FSM of M.
Proposition (3.26) A fuzzy submodule m of Z-module is G-invariant fuzzysubmodule of M if and only if mG = m.
Proof: This follows from Proposition (3.4) and Proposition (3.24).
Proposition (3.27) Let m be fuzzy submodule of Z-module M. Then mG is the largest G-invariant fuzzy submodule of M contained in m.
Proof: This follows immediately from Proposition (3.4) and Proposition (3.25)
Theorem (3.28) If mand n are G-invariant fuzzy submodules of M, then m + n is also a G-invariant fuzzy submodule on M.
Proof. Let x, yÎ M, gÎG be any elements, then
![]()
Hence m + n is a G-invariant fuzzy submodule of M.
Theorem (3.29) If m and n are G-invariant fuzzy submodules of M, then mn is also a G-invariant fuzzy submodule on M.
Proof: Let x, yÎ M, gÎG be any elements, then
![]()
Hence
is a G-invariant fuzzy submodule of M.
Definition (3.30) A non-constant fuzzy prime submodule g of Z-module M is called an G-invariant fuzzy G-prime submodule of M if for any two G-invariant fuzzy submodules m and nof M such that mn Í g implies that either m Í g or n Í g.
Theorem (3.31) If g is a fuzzy prime submodule of a Z-module M, then gG is G-invariant fuzzy G-prime submodule of M.
Proof: Let g be FPSM of M and let m and nbe two G-invariant FSM of M such that mn Í gG. Then mn Í g because gG Í g. Since g is a fuzzy prime submodule of M, either m Í g or n Í g, thus either m Í gG or n Í gG. Since gG is the largest G-invariant fuzzy submodule of M contained in g. Hence gG is G-invariant fuzzy G-prime submodule of M.
4. Homomorphism of Fuzzy G-Submodules
In this section, we study the image and pre image of fuzzy G-submodules under the module homomorphism.
Lemma (4.1) Let M and M¢ be Z-modules and G be a finite group which acts on M and M¢. Let f : M ® M' be a mapping defined by
, "x Î M, g Î G. Then f is a module homomorphism. We call the map f as G-module homomorphism.
Proof: Let x, y Î M, g Î G and r Î Z be any elements, then we have
![]()
Also ![]()
Hence f is a module homomorphism.
Lemma (4.2) Let M and M¢ be Z-modules and G be a finite group which acts on M and M¢. Let f : M ® M' be a G-module homomorphism and m and n are the fuzzy subsets of M and M' respectively. Then
(i) ![]()
(ii) ![]()
Proof. (i) Let
and
be any element. Then
![]()
Hence ![]()
(ii) Let
and
be any element. Then
![]()
Hence ![]()
Theorem (4.3) Let M and M¢ be Z-modules on which G acts and let f be a G-module homomorphism from M into M¢. If n be a fuzzy G-submodule of M¢, then f−1(n) is a fuzzy G-submodule of M.
Proof: Let n be a fuzzy G-submodule of M¢. To show that f−1(n) is a fuzzy G-submodule of M. It is equivalent to showing that
is a fuzzy submodule of M for all gÎG. But, in view of Lemma (4.2) (i), it is sufficient to show that f−1(ng) is a fuzzy submodule of M for all g Î G. For x, yÎM, rÎZ and gÎG, we have
![]()
![]()
![]()
![]()
Theorem (4.4) Let M and M¢ be Z-modules on which G acts and let f be a bijective G-module homomorphism from M into M¢. If m is a fuzzy G-submodule of M, then f (m) is a fuzzy G-submodule of M¢.
Proof: Let m be a fuzzy G-submodule of M. To show that f(m) is a fuzzy G-submodule of M¢ is equivalent to showing that
is a fuzzy submodule of M¢ for all gÎG.
For, in view of Lemma (4.2) (ii), it is sufficient to show that f(mg) is a fuzzy submodule of M¢ for all g Î G.
![]()
Then for any
there exists unique x, yÎM such that
and
.
![]()
![]()
![]()
Theorem (4.5) Let M and M¢ be Z-modules on which G acts and let f be a G-module homomorphism from M into M¢. If n be a G-invariant fuzzy submodule of M¢, then f−1(n) is a G-invariant fuzzy submodule of M.
Proof. Since n is G-invariant fuzzy submodule of M¢. Therefore, ng = n, for all g Î G.
![]()
Theorem (4.6) Let M and M¢ be Z-modules on which G acts and let f be a bijective G-module homo- morphism from M into M¢. If m is a G-invariant fuzzy submodule of M, then f (m) is a G-invariant fuzzy submodule of M¢.
Proof. Since m is G-invariant fuzzy submodule of M¢. Therefore, mg = m , for all gÎG.
![]()
5. Conclusion
In this paper, the notion of fuzzy G-submodule of a Z-module is defined and discussed. It has been proved that every fuzzy G-module is a fuzzy module but the converse is not true in general. It has also been proved that intersection and Cartesian product of two fuzzy G-submodules are fuzzy G-submodules. The notions of fuzzy prime submodule, fuzzy G-prime submodule, fuzzy semi prime submodule and fuzzy G-semi prime submodule are introduced and discussed. We have observed that intersection of two fuzzy prime submodules needs not be a fuzzy prime submodule; however intersection of two fuzzy prime submodules is always a fuzzy semi prime submodule. The notions of G-invariant fuzzy subset (submodule) and G-invariant fuzzy prime (G-prime) submodule of Z-module are also introduced and discussed. We have proved that sum and product of two G-inva- riant fuzzy sub-modules are G-invariant fuzzy submodules.