Some Properties of the Essential Fuzzy and Closed Fuzzy Submodules

In this paper, we introduce and study the essential and closed fuzzy submodules of a fuzzy module X as a generalization of the notions of essential and closed submodules. We prove many basic properties of both concepts.


Introduction
The notion of a fuzzy subset of a nonempty set S as a function from S into [0,1] was first developed by Zadeh [1]. The concept of fuzzy modules was introduced by Zahedi [2], whereas that of fuzzy submodules was introduced by Martines [3].
A non-zero proper submodule A of a module M is called an essential if A , for any nonzero submodule B of M [4,5].
Rabi [6] fuzzified this concept to the essential fuzzy submodule of a fuzzy module X. Goodeal [4,7] introduced and studied the concept of closed submodules, where a submodule A of an R-module M is said to be a closed submodule of M (A ), if it has no proper essential extension. In this paper, we shall give some properties of the essential fuzzy submodules. Also, we introduce the concept of the closed fuzzy submodules. We establish many properties and characterizations of these concepts.
Throughout this paper, R is a commutative ring with unity, M is an R-module, and X is a fuzzy module of an R-module M.

1.Preliminaries
In this section, we shall provide the concepts of fuzzy sets and operations on fuzzy sets, with some of their important properties which are used in the next sections of the paper.

ISSN: 0067-2904 Definition 1.1:[1]
Let M be non-empty set and let I be the closed interval Now, we give the definition of the level subset, which is a set between a fuzzy set and an ordinary set.

Definition 1.5:[3]
Let A be a fuzzy in M, for all t [ ], then the set { } is called level subset of X.
Note that, is a subset of M in the ordinary sense. The following are some properties of the level subset. Remark 1. 3. X(0) = 1 (0 is the zero element of M).

Definition 1.9: [3]
Let X and A be two fuzzy modules of an R-module M. A is called a fuzzy submodule of X if A X. Proposition 1.10: [8] Let A be a fuzzy set of an R-module M. Then the level subset , t [0,1] is a submodule of M if and only if A is a fuzzy submodule of X, where X is a fuzzy module of an R-module M. Now, we shall give some properties of the fuzzy submodule, which are used in the next section. Definition 1.11: [2] Let A and B be two fuzzy subsets of an R-module M. Then Let A and B be two fuzzy submodule of a fuzzy module X, then A+B is a fuzzy submodule of X. Remark 1.13: [9] If X is a fuzzy module of an R-module M and X, then for all fuzzy singleton of R , = min{ }. Definition 1.14: [6] Let X and Y be two fuzzy modules of an R-module and respectively . Define X Y : { } is called support of A, also , t ].

2.
{ } Definition 1.16: [10] A fuzzy module X of an R-module M is called simple if X = . Definition 1.17: [6] Let A be a fuzzy submodule of a fuzzy module X of an R-module M, then A is called an essential fuzzy submodule (briefly A ), if A B , for any non-trivial fuzzy submodule B of X. Equivalently, a fuzzy submodule A of X is called essential if A , implies that U = , for all fuzzy submodule U of X.

Properties of Essential Fuzzy Submodules
In this section we shall give some properties of essential fuzzy submodule which were introduced in a previous study [6].

Remark 2.1:
If X is a fuzzy module of an R-module M and A is an essential fuzzy submodule of X, then is not necessarily an essential of .  ]. Thus, B = ; that is A is an essential fuzzy submodule of X . The following condition remark will be needed in some properties in our work. Remark Let X be a fuzzy module of an R-module M and A, B are non-trivial fuzzy submodules of X, if implies that A B.

Proposition 2.3:
Let X be a fuzzy module of an R-module M such that X satisfies the previous remark and A be a fuzzy submodule of X, then A is an essential fuzzy submodule of X if and only if is an essential submodule in . Proof: ( ) Let be an essential submodule of . To prove that A is essential in X. Suppose that B is a fuzzy submodule of X, B and A , then (0), by remark(1.6)(3). So implies that , so (0) , since is an essential in . Then , then by the previous remark , implies that A . Therefore, . Thus, B = and A is essential in X. ( ) If A is essential in X, then to show that is essential in , let N be a submodule of and It is clear that B is a fuzzy submodule of X and , then . Thus, . Then by the previous remark, A . Since A is an essential fuzzy submodule of X, then (0), so implies that , by remark (1.6)(1) so (0), since is an essential in . Thus and so by the previous remark , B = and so A is an essential in X.

Remark 2.4:
Every non-trivial fuzzy submodule of X is essential fuzzy itself. Proof: Let A be non-trivial fuzzy submodule of X and B be fuzzy submodule of A, where B , then A B = B . Therefore, A is an essential fuzzy submodule of A.

Proposition 2.5:
Let X be a fuzzy module of an R-module M and , , , be fuzzy submodules of X, where is an essential fuzzy in and is an essential in , then is essential in . Proof: Let C be any fuzzy submodule of X such that . Since is essential in , then . Therefore, is non-trivial fuzzy submodule of X such that . As is essential in , hence , then . Therefore, is essential in . Proposition 2.6: Let X be a fuzzy module of an R-module M and be fuzzy submodules of X . If is essential in X and is essential in X, then is essential in X. Proof: It is clear by remark(2.4) and proposition(2.5).

Proposition 2.7:
Let X be a fuzzy module of an R-module M and A B X. If A is an essential fuzzy in B and B is an essential fuzzy in X, then A is an essential fuzzy submodule in X.

Proof:
Let A be an essential in B, B be an essential in X, and C be any non-trivial fuzzy submodule of X. Since B is an essential fuzzy submodule in X, then we have C B and then since A is essential in B, we have (C B) A ; that is C A . Thus, A is an essential fuzzy submodule in X. Recall that a submodule A of an R-module C. A relative complement for A in C is any submodule B of C which is maximal with respect to the property A [4]. We fuzzify this concept as follows: Definition 2.8: Let X be a fuzzy module of an R-module M and A be a fuzzy submodule of X. A relative complement for A in X is any fuzzy submodule B of X which is maximal with respect to the property A .

Proposition 2.9:
Let A be a fuzzy submodule of X of an R-module M, such that X satisfies the previous remark. If B is a relative complement for A, where B is any fuzzy submodule of X, then A is essential in X.

Proof:
Since B is a relative complement of A, we can prove that is a relative complement of . As B , then , so (0). Suppose that N is a submodule of and is a submodule of N such that N (0). Let C: M [ ], define by: It is clear that C is a fuzzy submodule of X and . Thus = (0) and so . By the previous remark, C A = . But is a submodule of = N, implies that B is a fuzzy submodule of C (by the previous remark). Thus C = B, since B is a relative complement of A. It follows that (see remark (1.6)(3)); that is N = and is a relative complement of . Hence, is an essential submodule in [4,proposition(1. 3)]. So is an essential submodule in . Therefore, A B is an essential fuzzy submodule of X by proposition (2.4).

3.Properties of Closed Fuzzy Submodules
In this section, we introduce the notion of the closed fuzzy submodule of a fuzzy module as a generalization of (ordinary) notion closed submodule, where a submodule A of an R-module M is said to be closed submodule of M (briefly A , if A has no proper essential extension ; that is if A , then A = B [4], [7]. We shall give some properties of this concept. is closed fuzzy submodule of X and X is closed fuzzy submodule of X .

Proof:
Let A be a direct summand of X , then there exists fuzzy submodule of X such that X = A then A + B = X and A . Suppose that A is an essential fuzzy submodule of C , where C is a fuzzy submodule of X . To show that A = C, we claim that B . Since , then A . If B , then A , since B and A is essential in C, which is a contradiction. Then B . Therefore, B + C = B C. Now, X = A , since A . Then X = A , so A , since .
, then by a previous study [6,

If A is closed fuzzy submodule of X of an R-module M and A
, then A is a closed fuzzy in B.

Proof:
Assume that A is essential in D, where D is a fuzzy submodule of B .It is clear that D is a fuzzy submodule of X . Hence, A = D, since A is a closed fuzzy in X . Thus A is a closed fuzzy in B.

Theorem 3.4:
Let { } and { } be collections of a fuzzy modules of an R-module M, such that is a closed fuzzy submodule of , for each .
It is clear that A is a fuzzy submodule of X and N, so is a relative complement of . Hence, and so B A = . Suppose that C is a fuzzy submodule of X and B is a fuzzy submodule of C such that C , so . But is a relative complement of , so . Thus B = C (by the previous remark) and B is a relative complement of A.
(2) (1) : If B is a relative complement of A. To prove that B is a closed fuzzy submodule in X, suppose that B . Then B , by proposition (2.6). Hence, and so C But B is a fuzzy submodule of C and B is a relative complement of A, hence C = B. Thus, B is closed fuzzy submodule in X. ( Let X be a fuzzy module of an R-module M and let A . If A is closed fuzzy in B and is B closed fuzzy in X, then A is closed fuzzy in X. Proof: Since A is closed fuzzy in B and B is closed fuzzy in X, therefore is closed in and is closed in by proposition (3.2). This implies that is closed in [4 , proposition(1.5)]. Thus, A is closed fuzzy in X by proposition (3.2

Remark 3.8:
The intersection of two closed fuzzy submodules needs not be a closed fuzzy submodule in general, as the following example shows: Example It is clear that A and B are fuzzy submodules of X and that ( ) , ( ) . Since is a direct summand of , then A is direct summand of X, by proposition (3.7). Also is direct summand of , then B is direct summand of X . Therefore, A and B are closed fuzzy submodules of X, by remark and example (3.3)(2). But (A min { } (A { ( ) ( ) , which is not a closed submodule in , since (2, ( ) , but (2, ( ) Therefore, A is not a closed fuzzy submodule of X, by proposition (3.2).