RP-T-fuzzy soft subrings and ideals of soft rings

In this paper we introduce a concept which is called RP - T -fuzzy soft subring and examine some properties of the restricted intersection, the restricted union, the ∧ -intersection and the product of their families. A condition to make the restricted union of RP - T -fuzzy soft subrings to be RP - T -fuzzy soft subring is determined. A correlation between the RP - T -fuzzy soft subring of a soft ring and α -level sets of this soft ring is demonstrated. The RP - T -fuzzy soft subrings under some binary operations are investigated. Moreover, the image and pre-image of RP - T -fuzzy soft subrings under fuzzy soft homomorphisms is examined. Finally, we present the concept of RP - T -fuzzy soft ideal and we investigate the analogue properties for them.


Introduction
Most of the notions in all fields of the real world have uncertainties and vagueness. In 1965, Zadeh initiated fuzzy sets (or fuzzy subsets) as a class of objects with a continuum of grades of membership to deal with uncertain concepts [1]. Many researchers have established its connection with almost every topic since fuzzy sets introduced. The fuzzy sets were applied to algebra by presenting fuzzy groups by Rosenfeld [2] in 1971. The notion of fuzzy subrings was introduced by Liu in 1982 [3]. Dixit et. al. provide an internal description of the fuzzy subring and fuzzy ideal generated by a finite fuzzy subset of a ring [4].
In 1999, Molodtsov introduced the theory of the soft set has a large area of use for solving uncertain problems [5]. Maji et. al. put forth some basic algebraic features such as equality, union and intersection, null and absolute of two soft sets, and also its complement [6]. Ali et. al. discuss some situations at Maji et. al's study and give some new notions [7]. The theory of soft sets can be combined with some other theories. In 2007, Aktaş and Çağman combined soft sets with algebraic concepts and they defined the notions of soft group, soft subgroup, soft normal subgroup and soft homomorphism of soft sets [8]. Acar et. al. [9] introduce initial concepts of soft rings. Atagün and Sezgin [10] study the algebraic soft substructures of rings, fields and modules.
In 2001, Maji et. al. amalgamated soft and fuzzy sets and they defined fuzzy soft sets [11]. In 2009, Aygünoğlu and Aygün introduced a fuzzy soft group as a new concept. Defined fuzzy soft function, they also introduced a fuzzy soft homomorphism of fuzzy soft groups. Moreover, they gave the concept of normal fuzzy soft group and they investigated some of its basic properties [12]. Pazar Varol et. al. [13] introduce the concept of a fuzzy soft ring and study some of their algebraic features. İnan and Öztürk [14] present the notion of a fuzzy soft ring and (ε, ε ∧ q)-fuzzy soft subring that is a generalization of the fuzzy soft ring. Çelik et. al. [15] provide a fuzzy extension of soft rings introduced by Acar et al [9]. Some recent papers show that investigations related to the theory of soft set continue rapidly [16][17][18][19][20][21][22][23][24][25][26].
Akın and Karakaya [16] propose new algebraic notion which is called UP-fuzzy soft subset of a soft set, where U denotes a universal set and P denotes a set of parameters. They present the notions SP-fuzzy soft semigroup and SP-fuzzy soft left (right) ideal of a soft SP-fuzzy soft semigroup. Then, Akın [17] define GP-fuzzy soft groups. In this paper, the concept of an RP-T-fuzzy soft subring is introduced. The concept of RP-T-fuzzy soft subring, where T is a t-norm, is a combination of the notion of UP-fuzzy soft subset and ring theory. Some properties of the restricted intersection, ∧-intersection and product of their families are studied. It is demonstrated that the restricted union of RP-T-fuzzy soft subrings of a soft ring is an RP-T-fuzzy soft subring of this soft ring if T is an infinitely ∨-distributive t-norm. Relations of the RP-T-fuzzy soft subring of a soft ring and α-level sets of this soft ring are investigated. Some RP-T-fuzzy soft subrings which are obtained by some binary operations are examined. Moreover, the image and pre-image of RP-T-fuzzy soft subrings under a fuzzy soft homomorphism is investigated. Finally, this paper presents the concept of RP-T-fuzzy soft ideal and gives some analogue properties for them.

Fuzzy subsets
Let U be a universe of discourse. A function f from U to [0,1] is called a fuzzy subset of U. The family of all fuzzy subsets of U is denoted by ℱ(U). Let f, g ∈ ℱ(U). Then, f ⊆ g means that f(a) ≤ g(a) for all a ∈ U. f * g is a binary relation on ℱ(U) defined by (f * g)(u) = f(u) * g(u) for all u ∈ U, where " * " is a binary relation on [1,27]).

Subrings, ideals, -fuzzy subrings and -fuzzy ideals
Let R denote a commutative ring and ∅ ≠ I ⊆ R. Then, I is called a subring of R if a − b, a. b ∈ I for all a, b ∈ I. A subring I is a left ideal if ra ∈ I for all a ∈ I, r ∈ R. Let f ∈ ℱ(R) and T be a t-norm. Then, f is called a T-fuzzy for all x, y ∈ R. f is called a T-fuzzy ideal (or two-sided ideal) if its both T-fuzzy left and right ideal (See [27,33,34]).

Soft sets and soft rings.
In this part, we present some known definitions of topics of soft sets and soft rings.
Definition 1 [5] Let U be an initial universe set and P be a set of parameters. The power set of U is denoted by P(U) and A is a subset of P. A pair (F, A) is called a soft set over U where F is a mapping given by F: A → P(U). The pair (U, P) denotes the collection of all soft sets on U with the attributes from P and is called a soft class [35].
Definition 2 [5] Let (F, A) and (G, B) be two soft sets over U , (F, A) is called a soft subset of (G, B), denoted by Definition 3 [6] Let (F, A) be soft set over U. Then, (ii) (F, A) is said to be an absolute soft set, denoted by Ã, if F(x) = U for all x ∈ A.
Definition 4 [36] For a soft set (F, A), the set Supp(F, A) = {x ∈ A|F(x) ≠ ∅} is called the support of the subset (F, A). If Supp(F, A) ≠ ∅, then the soft set (F, A) is called non-null.

Fuzzy soft sets and Fuzzy soft rings
In this part, we give some known and useful definitions of topics of fuzzy soft sets and fuzzy soft rings.
Definition 7 [11] Let U be an initial universe set and P be a set of parameters. A pair (f, E) is called a fuzzy soft set over U where f: E → F(U) is a mapping. The pair (U, P) denotes the collection of all fuzzy soft sets on U with the attributes from P and is called a fuzzy soft class (See [40]).
Definition 9 [11] Let (f, E) and (g, H) be two fuzzy soft sets over U , (f, E) is called a fuzzy soft subset of (g, H), denoted by (f, E) ⊆ (g, H), if (i)E ⊆ H, (ii) for each a ∈ E, f(a) ≤ g(a). [11,15,40] Let {(f i , E i )|i ∈ Λ} be a family of fuzzy soft sets in a fuzzy soft class (U, P). Then,

Definition 10
Definition 11 [11,15] Let {(f i , E i )|i ∈ Λ} be a family of fuzzy soft sets in a fuzzy soft class (U, P) . Then, Definition 12 [15] Suppose that ⊕ is a binary operation on the power set of P and ⊗ is a binary operation on F(U). Then, for any two fuzzy soft sets (f, E), (g, A) ∈ (R, P) , (f, E) ⊕ ⊗ (g, A) is defined as the Φ-fuzzy soft set (h, C), where C = E ⊕ A and for all x ∈ C, where Φ is arbitrary fuzzy set of U. Clearly," ⊕ ⊗ " is a binary operation on (R, P) .
Definition 13 [15] Let (f, E 1 ), (g, E 2 ) be fuzzy soft sets in a fuzzy soft class (U, P) . Then, the fuzzy product of them, denoted by (f, Definition 14 [14,15] Let (f, E) be a fuzzy soft set over R. Then, Definition 15 [14] Let (f, A) be a fuzzy soft ring over R. A fuzzy soft set (g, B) over R is called a fuzzy soft ideal of (f, A) if and only if Definition 16 [12] Let (f, A) and (g, B) fuzzy soft sets on the classes (U 1 , P 1 ) and (U 2 , P 2 ) , respectively and let φ: U 1 → U 2 , ψ: A → B be functions. Then, the pair (φ, ψ) is called a fuzzy soft function from U 1 to U 2 .
Definition 17 [12] Let (f, A) and (g, B) fuzzy soft sets on the classes (U 1 , P 1 ) and (U 2 , P 2 ) , respectively and let (φ, ψ) be a fuzzy soft function from U 1 to U 2 . Then, , is the fuzzy soft set on the class (U 2 , P 2 ) defined by Definition 18 [13] Let (φ, ψ) be a fuzzy soft function from R to S. If φ is a ring homomorphism from R to S, then (φ, ψ) is said to be a fuzzy soft homomorphism. If φ is an isomorphism and ψ is a one-to-one mapping from A onto B, then (φ, ψ) is said to be a fuzzy soft isomorphism. Definition 20 [16] Let (f, E) be a UP-fuzzy soft subset of (F, A). (F α , E) called α-level soft subset of (f, E), where F α : E → P(U) is defined by F α (x) = {α ∈ F(x)|f(x)(a) ≥ a} for all x ∈ E.
Definition 21 [16] Let (f, E) be a UP-fuzzy soft subset of (F, A) . Then, the UP-fuzzy soft subset (f, E) c = (f c , E) of (F, A) is called the complement of (f, E), where for any x ∈ E,f c (x): Theorem 22 [16] Let {(f i , E i )|i ∈ Λ} be the family of UP-fuzzy soft subset of (F, A). Then, De Morgan rules are provided for restricted intersection and union, i.e.,

Results
Throughout this paper, all fuzzy subsets are considered over commutative rings R and Z. T, S and I denote a tnorm, a t-conorm and an implication on [0,1], respectively, unless otherwise stated. We begin by giving a definition of RP-T-fuzzy soft subring.
Proposition 26 Let (f, E) be an RP-T-fuzzy soft subring of (F, A). If (F α , E) is a soft left or right ideal over R, then it is a soft ring over R.

Proof. It is straightforward.
Theorem 27 Let (f i , E i ) be an RP-T-fuzzy soft subring of (F i , A i ) for all i ∈ Λ. Then, . Hence, f(x) is RP-T-fuzzy soft subring of F(x) for all x ∈ E. Thus, ⋂ i∈Λ (f i , E i ) is RP-T-fuzzy ring of (⋂ r ) i∈Λ (F i , A i ).
ii) The proof is similar to the proof of Theorem 3.4 (b) in [17] with the definitions T-fuzzy subring and RP-Tfuzzy soft subring.
iii) The proof is similar to the proof of Theorem 3.4 (c) in [17] with the definitions T-fuzzy subring and RP-Tfuzzy soft subring. iv) Let ∧ i∈Λ (f i , E i ) = (f, E) and ∧ i∈Λ (F i , A i ) = (F, A). Let ((x i ) i∈Λ ) ∈ E and a, b ∈ F((x i ) i∈Λ ). Then, Therefore, ∏ i∈Λ (f i , E i ) is an RP-T-fuzzy soft subring of ∏ i∈Λ (F i , A i ) since f((x i ) i∈Λ ) is a fuzzy subring of F((x i ) i∈Λ ).

Theorem 28
Let {(f i , E i )|i ∈ Λ} be a family of RP-T-fuzzy soft subrings of (F, A). Then, ⋃ i∈Λ (f i , E i ) is an RP-T-fuzzy soft subring of (F, A) if T is an infinitely ∨-distributive t-norm.
Thus, f(x) is a T-fuzzy subring of F(A). Therefore, ⋃ i∈Λ (f i , E i ) is an RP-T-fuzzy soft subring of (F, A).
. Therefore, (f, E) ⋂ T (g, A) is an RP-T-fuzzy soft subring of (F, K) ⋂ r (G, L).  Proof. Let y 1 , y 2 ∈ φ(F)(b) for any b ∈ ψ(E). Suppose that there exist x 1 , x 2 ∈ R such that φ(x 1 ) = y 1 and φ(x 2 ) = y 2 . Then, the inequality is verified by the similar way in the proof of Theorem 3.8 in [17] with the use of the infinitely ∨-distributivity of T. Thus, the proof of Theorem 3.8 in [17] completes the proof.

Theorem 40
Let {(f i , A i )|i ∈ Λ} be a family of the RP-T-fuzzy soft left (right) ideals of (F i , A i ) for all i ∈ Λ.
Proof. It is straightforward.

Theorem 41
Let {(f i , E i )|i ∈ Λ} be a family of RP-T-fuzzy soft left (right) ideals of (F, A). Then ⋃ i∈Λ (f i , E i ) is an RP-T-fuzzy soft left (right) ideal of (F, A) if T is an infinitely ∨-distributive t-norm.
Therefore, ⋃ i∈Λ (f i , E i ) is an RP-T-fuzzy soft left ideal of (F, A) by Theorem 28. The proof is similar for RP-Tfuzzy soft right ideals.
Proof. Let (f, E) ∪ T (g, A) = (h, C) and (F, K) ⋂ e (G, L) = (H, B). Then, C = E ∪ A ⊆ K ∪ L = B. The rest of the steps are similar to the proof of Theorem 42.