NOTE ON SOFT FRACTIONAL IDEAL OF RING

In this note we introduce soft fractional ideal of soft rings. Then, we study fractional ideal by applying few basic soft operations.


Introduction and Preliminaries
Theory of probability, theory of fuzzy sets [13], theory of intuitionistic fuzzy sets [4], theory of vague sets [7], theory of interval mathematics [8], and theory of rough sets [11] which were considered best mathematical tools for dealing with uncertainties.In [10], Molodtsov showed that to fix uncertainties soft set theory works more efficient than any other tool.In [2] authors discussed soft groups, soft subgroups.In [1] soft rings, soft ideals of soft rings have been introduced, furthermore the authors also introduced idealistic soft rings.For basic terminologies of soft set one may consult [10] and, for soft rings and soft ideals we refer [1].In the begining we recall few useful definitions and terminologies.
Let R be an integral domain, and K be its field of fractions.R-submodule I of K such that there exists a non-zero r ∈ R such that rI ⊆ R is said to be a fractional ideal of R. Every integral ideal is a fractional ideal of ring R.This type of ideal has its own importance while study Dedekind domains, valuation, domains etc. Following [10, definition 2.1] pair (F, E) is called a soft set (over U) if and only if F is a mapping on E into the set of subsets of the set U. Assume that (F, A) and (H, B) are two soft sets over a common universe U .We say that (F, A) is a soft subset of (H, B), if it satisfies: (1) A ⊂ B and (2) F(x) and H(x) are identical approximations for all x ∈ A [10].In [1, definition 3.1] authors introduced soft rings i.e., Let (F, A) be a non-null soft set over a ring R. Then (F, A) introduce soft ideal of a soft ring i.e., Let (F, A) is a soft ring over R. A non-null soft set (γ, I) over R is called soft ideal of (F, A), if it satisfies: (1) I ⊂ A and (2) γ(x) is an ideal of F(x) for all x ∈ Supp(γ, I).Throughout this paper E is a set of parameters, P(R) is the power set of R , Z is the ring of integer numbers.Definition 1.Let (F, A) and (G, B) be two soft sets over a common universe U, we say that and (G, B) be two soft sets over a common universe U.The intersection of (F, A) and (G, B) is defined as the soft set (H,C) satisfying the following conditions: or G(x) (while the two sets are the same).
In this case we write Definition 3. Let (F, A) and (G, B) be two soft sets over a common universe U.The biintersection of (F, A) and (G, B) is defined as the soft set (H,C) satisfying the following conditions: In this case we write H(C) = F(A) ∼ G(B) [9].Definition 4. Let (F, A) and (G, B) be two soft sets over a common universe U.The union of (F, A) and (G, B) is defined as the soft set (H,C) satisfying the following conditions: In this case we write A soft set (F, A) is said to be non-null if its support is not equal to empty set [6].

Soft fractional ideal of rings
Fuzzy fractionary ideal has been introduced and discussed in the literature(see [5]).Different types of soft ideals have been also introduced in the literature.Soft substructures of rings, fields and modules have been discussed in the literature [3].Soft module and submodules have been introduced in the literature [12].In this section we introduce and discuss about soft fractional ideals of soft rings.Throughout by R we mean an integral domain and K be its field of fraction.
We begin with the definition.
Definition 13.Let µ be a soft set over the field K and µ α = {x ∈ K : µ(x) ⊇ α} be a level set for every α ∈ P(K).
We let χ A the characteristic function for a subset A of a ring R ⊆ K. Let χ α A be a soft subset of K such that χ α A (x) = U, if x ∈ R, and χ α A (x) = α if x ∈ K − R, where α ∈ P(K).A soft subset µ is said to be a soft ideal of a ring R if µ(x − y) ⊇ µ(x) ∩ µ(y) and µ(xy) ⊇ µ(x) ∪ µ(y).A soft subset of R is said to be an ideal iff µ(0) ⊇ µ(x) for every x ∈ R and µ α is an ideal for every α ∈ P(K).Definition 14.Let R be a ring contained in a field K, and (β , K) be a soft subset over the field K. Then β is said to be soft R-submodule of K if: definition 2.11].Definition 5.If (F, A) and (G, B) be two soft sets over a common universe U. Then "(F, A) AND (G, B)" denoted by F(A) ∼ ∧ G(B) is defined as F(A) ∼ ∧ G(B) = (H, C), where C = A × B and H(x, y) = F(x) ∩ G(y) for all (x, y) ∈ C [9, definition 2.9].Definition 6.If (F, A) and (G, B) be two soft sets over a common universe U. Then "(F, A) OR (G, B)" denoted by F(A) ∼ ∨ G(B) is defined as F(A) ∼ ∨ G(B) = (H, C), where C = A × B and H(x, y) = F(x) ∩ G(y) for all (x, y) ∈ C [9, definition 2.10].Definition 7. Let (F, A) be a soft set.The support of (F, A) i.e., Supp(F, A) = {x ∈ A| F(x) = / 0}.

Definition 8 .Definition 9 .( 1 )
Let (F, A) be a non-null soft set over a ring R.Then (F, A) is called a soft ring over R if F(x) is a subring of R for all x ∈ A [1, definition 3.1].Let (F,A) is a soft ring over R, a non-null soft set (γ, I) over R is called soft ideal of (F, A), and is denoted by (γ, I)∼ (F, A) if it satisfies: I ⊂ A (2) γ(x) is an ideal of F(x)for all x ∈ Supp(γ, I) [1, definition 4.1].Definition 10.Let (F, A) and (G, B) be non-null soft sets over a ring R. Then (G, B) is called a soft subring of (F, A) if it satisfy the following (1) A ⊂ B (2) G(x) is a subring of F(x), for all x ∈ Supp(G, B) [1, definition 4.1].Definition 11.Let (F, A) be a non-null soft sets over a ring R. Then (F, A) is called an idealistic soft ring over R, if F(x) is an ideal of R for all x ∈ Supp(F, A) [1, definition 5.1].Definition 12. Let M be a left R-module, A be any nonempty set F : A → P(M) refers to a set-valued function and the pair (F, A) is a soft set over M.Then, (F, A) is said to be a soft module over M if and only if F(x) < M for all x ∈ A [12].