Intuitionistic L-Fuzzy Rings. By K. Meena & K. V. Thomas Bharata Mata College, Thrikkakara

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1 Global Journal of Science Frontier Research Mathematics and Decision Sciences Volume 12 Issue 14 Version 1.0 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA Online ISSN: & Print ISSN: Intuitionistic L-Fuzzy Rings By K. Meena & K. V. Thomas Bharata Mata College, Thrikkakara Abstract - In this paper we study some generalized properties of Intuitionistic L-fuzzy subrings. In this direction the concept of image and inverse image of an Intuitionistic L-fuzzy set under ring homomorphism are discussed. Further the concept of Intuitionistic L-fuzzy quotient subring and Intuitionistic L-fuzzy ideal of an Intuitionistic L-fuzzy subring are studied. Finally, weak homomorphism, weak isomorphism, homomorphism and isomorphism of an Intuitionistic L-fuzzy subring are introduced and some results are established in this direction. Keywords : intuitionistic l-fuzzy quotient subring, intuitionistic l-fuzzy ideal of an intuitionistic l- fuzzy subring. GJSFR-F Classification : MSC 2010: 16P70 Intuitionistic L-Fuzzy Rings Strictly as per the compliance and regulations of :. K. Meena & K. V. Thomas. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Ref. Intuitionistic L-Fuzzy Rings K. Meena α & K. V. Thomas σ [1] Adam Niewiadomski and Eulalia Szmidt, Handling Uncertainty in natural sentences via IFS, on Intuitionistic Fuzzy Sets, 7 (2001, No.1, Abstract - In this paper we study some generalized properties of Intuitionistic L-fuzzy subrings. In this direction the concept of image and inverse image of an Intuitionistic L-fuzzy set under ring homomorphism are discussed. Further the concept of Intuitionistic L-fuzzy quotient subring and Intuitionistic L-fuzzy ideal of an Intuitionistic L-fuzzy subring are studied. Finally, weak homomorphism, weak isomorphism, homomorphism and isomorphism of an Intuitionistic L-fuzzy subring are introduced and some results are established in this direction. Keywords : intuitionistic l-fuzzy quotient subring, intuitionistic l-fuzzy ideal of an intuitionistic l-fuzzy subring. The theory of Intuitionistic fuzzy sets plays an important role in modern mathematics. The idea of Intuitionistic L-fuzzy set (ILFS was introduced by Atanassov (1986 [3 5] as a generalisation of Zadeh s (1965 [15] fuzzy sets. Rosenfeld (1971 [14] has applied the concept of fuzzy sets to the theory of groups. Many researchers [1, 2, 6 10] applied the notion of Intuitionistic fuzzy concepts to set theory, relation, group theory, topological space, knowledge engineering, natural language, neural network etc. This paper is a continuation of our earlier paper [13]. Along with some basic results, we introduce and study Intuitionistic fuzzy quotient subrings and Intuitionistic fuzzy ideal of an Intuitionistic fuzzy subring of a ring. Further we define homomorphism and isomorphism of Intuitionistic fuzzy subrings of any two rings. Using this we establish the fundamental theorem of ring homomorphism and the third isomorphism theorem of rings for Intuitionistic fuzzy subrings. Infact, we emphasize the truth of the results relating to the non-membership function of an Intuitionistic fuzzy subring on a lattice (L,,,. The proof of the results on the membership function of Intuitionistic fuzzy subrings, Intuitionistic fuzzy ideals of an Intuitionistic fuzzy subring and Intuitionistic fuzzy Quotient ring are omitted to avoid repetitions which are already done by researchers Malik D.S and Mordeson J.N. [11, 12]. 17 In this section we list some basic concepts and well known results of Intuitionistic L-fuzzy sets, Intuitionistic L-fuzzy subrings and Intuitionistic L-fuzzy ideals [13]. Author α σ : P.G. Research Centre in Mathematics, Bharata Mata College, Thrikkakara. Global Journals Inc. (US

3 Throughout this paper (L,,, denotes a complete distributive lattice with maximal element 1 and minimal element 0, respectively. Let R and S be commutative rings with binary operations + and. Definition 2.1. Let X be a non-empty set. µ : X L. A L-fuzzy set µ of X is a function Definition 2.2. Let (L, be the lattice with an involutive order reversing operation N : L L. Let X be a non-empty set. An Intuitionistic L-fuzzy set (ILFS A in X is defined as an object of the form A = { x, µ A (x, ν A (x /x X} 18 where µ A : X L and ν A : X L define the degree of membership and the degree of non membership for every x X satisfying µ A (x N(ν A (x. Definition 2.3. Let A = { x, µ A (x, ν A (x /x X} and B = { x, µ B (x, ν B (x /x X} be two Intuitionistic L-fuzzy sets of X. Then we define (i A B iff for all x X, µ A (x µ B (x and ν A (x ν B (x (ii A = B iff for all x X, µ A (x = µ B (x and ν A (x = ν B (x (iii A B = { x, (µ A µ B (x, (ν A ν B (x /x X} where µ A µ B = µ A µ B, ν A ν B = ν A ν B (iv A B = { x, (µ A µ B (x, (ν A ν B (x /x X} where µ A µ B = µ A µ B, ν A ν B = ν A ν B. Definition 2.4. An Intuitionistic L-fuzzy subset A = { x, µ A (x, ν A (x /x R} of R is said to be an Intuitionistic L-fuzzy subring of R (ILFSR if for all x, y R, (i µ A (x y µ A (x µ A (y (ii µ A (xy µ A (x µ A (y (iii ν A (x y ν A (x ν A (y (iv ν A (xy ν A (x ν A (y. Proposition 2.5. If A = { x, µ A (x, ν A (x /x R} is an ILFSR. Then (i µ A (0 µ A (x and ν A (0 ν A (x for all x R (ii if R is a ring with identity 1 then µ A (1 µ A (x and ν A (1 ν A (x, for all x R. Theorem 2.6. Let A = { x, µ A (x, ν A (x /x R} and B = { x, µ B (x, ν B (x /x R} be two ILFSR. Then A B is an ILFSR of R. Definition 2.7. Let A = { x, µ A (x, ν A (x /x R} be an ILFSR of R. Then A is called an Intuitionistic L-fuzzy ideal of R (ILFI if, Global Journals Inc. (US

4 (i µ A (x y µ A (x µ A (y (ii µ A (xy µ A (x (iii ν A (x y ν A (x ν A (y (iv ν A (xy ν A (x, for all x, y R. Definition 2.8. Let A = { x, µ A (x, ν A (x /x R} be an ILFI of R. Then we define (µ A = {x R/µ A (x = µ A (0} (ν A = {x R/ν A (x > ν A (0}. Proposition 2.9. Let A = { x, µ A (x, ν A (x /x R} be an ILFI. If µ A (x y = µ A (0 then µ A (x = µ A (y and if ν A (x y = ν A (0 then ν A (x = ν A (y, for all x, y R. Proposition Every ILFI is an ILFSR. Definition Let A = { x, µ A (x, ν A (x /x R} be an ILFSR. Let x R, then C = { x, (µ A (0 {x} + µ A (x, (ν A (0 {x} + ν A (x /x R} is called an Intuitionistic L-fuzzy coset (ILFC of A and is denoted as Definition C = { x, (x + µ A (x, (x + ν A (x /x R}. Let R/A={(x+μμ A,(x+ ν A /x belongs to R} Let R/A = {(x + µ A, x + ν A /x R}. Define + and on R/A by (i (x + µ A + (y + µ A = x + y + µ A, (ii (x + ν A + (y + ν A = x + y + ν A, for all x, y R, and (iii (x + µ A (y + µ A = xy + µ A, (iv (x + ν A (y + ν A = xy + ν A, for all x, y R. be an ILFI. Then R/A is a ring with respect to + and and is called Quotient ring of R by µ A and ν A. Here we define the image and inverse image of ILFS under ring homomorphism and study their elementary properties. Using this, Correspondence Theorem for ILFI is established. This section also provides a definition for an Intuitionistic L-fuzzy quotient subring of an ILFSR relative to an ordinary ideal of a ring. 19 Global Journals Inc. (US

5 Definition 3.1. Let f : R S be a ring homomorphism. Let A = { x, µ A (x, ν A (x /x R} and B = { x, µ B (x, ν B (x /x S} be ILFS. Then C = { y, f(µ A (y, f(ν A (y /y S} is called Intuitionistic Image of A, where {µ A (x/x R, f(x = y} if f 1 (y f(µ A (y = 0 otherwise {ν A (x/x R, f(x = y} if f 1 (y f(ν A (y = 0 otherwise 20 for all y S; and D = { x, f 1 (µ B (x, f 1 (ν B (x /x R} is called Intuitionistic Inverse Image of B, where f 1 (µ B (x = µ B (f(x f 1 (ν B (x = ν B (f(x for all x R. Here f(µ A and f(ν A are called the image of µ A and ν A under f. Also f 1 (µ B and f 1 (ν B are called the inverse image of µ B and ν B under f. The proof of the following result is direct. Lemma 3.2. Let f : R S be a ring homomorphism. Let A = { x, µ A (x, ν A (x /x R} and B = { x, µ B (x, ν B (x /x S} be ILFI of R and S respectively. Then (i f(ν A (0 = ν A (0 where 0 is the zero element of S and 0 is the zero element of R. (ii f(ν A (f(ν A ; (iii If ν A has the infimum property, then f(ν A = (f(ν A ; (iv If ν A is constant on Ker f, then (f(ν A (f(x = ν A (x for all x R. Theorem 3.3. Let f : R S be a ring homomorphism. Let A = { x, µ A (x, ν A (x /x R} and B = { x, µ B (x, ν B (x /x S} be ILFI of R and S. Then (i D = { x, f 1 (µ B (x, f 1 (ν B (x /x R} is an ILFI of R which is a constant on Ker f; (ii f 1 (ν B = (f 1 (ν B ; (iii If f is onto then (f f 1 (ν B = ν B ; (iv If ν A is constant on Ker f, then (f 1 f(ν A = ν A. Global Journals Inc. (US

6 Proof. (i Let x, y R. Then f 1 (ν B (x y = ν B (f(x y = ν B (f(x f(y ν B (f(x ν B (f(y = f 1 (ν B (x f 1 (ν B (y and f 1 (ν B (xy = ν B (f(xy = ν B (f(xf(y ν B (f(x ν B (f(y 21 Hence D is an ILFI of R. Let x Ker f. Then Hence f 1 (ν B is constant on Ker f. (ii Let x R. Then = f 1 (ν B (x f 1 (ν B (y. f 1 (ν B (x = ν B (f(x = ν B (f(0 = ν B (0. x f 1 (ν B ν B (f(x > ν B (0 = ν B (f(0 Hence f 1 (ν B = (f 1 (ν B. f 1 (ν B (x > f 1 (ν B (0 x (f 1 (ν B. (iii Let y S. Then y = f(x for some x R, so that (f f 1 (ν B (y = f(f 1 (ν B (y = f(f 1 (ν B (f(x = f 1 (ν B (x = ν B (f(x = ν B (y. Global Journals Inc. (US

7 22 Hence (f f 1 (ν B = ν B. (iv Let x R. Then (f 1 f(ν A (x = f 1 (f(ν A (x = f(ν A (f(x = ν A (x. Hence (f 1 f(ν A = ν A. Theorem 3.4. Let f : R S be an onto ring homomorphism. Let A = { x, µ A (x, ν A (x /x R} be an ILFI of R. Then C = { y, f(µ A (y, f(ν A (y /y S} is an ILFI of S. If ν A is a constant on Ker f, then f(ν A = (f(ν A. Proof. Let s 1, s 2 S. Then s 1 = f(r 1, s 2 = f(r 2 for some r 1, r 2 R. Now f(ν A (s 1 s 2 = {ν A (x/x R, f(x = s 1 s 2 } {ν A (r 1 r 2 /r 1, r 2 R, f(r 1 = s 1, f(r 2 = s 2 } {ν A (r 1 ν A (r 2 /r 1, r 2 R, f(r 1 = s 1, f(r 2 = s 2 } = ( {ν A (r 1 /r 1 R, f(r 1 = s 1 } ( {ν A (r 2 /r 2 R, f(r 2 = s 2 } = f(ν A (s 1 f(ν A (s 2. Also f(ν A (s 1 s 2 = {ν A (x/x R, f(x = s 1 s 2 } {ν A (r 1 r 2 /r 1, r 2 R, f(r 1 = s 1, f(r 2 = s 2 } {ν A (r 1 ν A (r 2 /r 1, r 2 R, f(r 1 = s 1, f(r 2 = s 2 } = ( {ν A (r 1 /r 1 R, f(r 1 = s 1 } ( {ν A (r 2 /r 2 R, f(r 2 = s 2 } = f(ν A (s 1 f(ν A (s 2. Hence C = { y, f(µ A (y, f(ν A (y /y S} is an ILFI(S. Next, if ν A is a constant on Ker f, then for y (f(ν A, we have f(ν A (y > f(ν A (0 = ν A (0. Since f is onto, y = f(x for some x R. Hence f(ν A (f(x = ν A (x > ν A (0. Thus ν A (x > ν A (0 or x (ν A. Hence y = f(x f(ν A. The remaining part of the proof follows from Lemma 3.2 (ii. Global Journals Inc. (US

8 Let R be a ring. Let R/A = {(x + µ A, (x + ν A /x R} be a quotient ring by µ A and ν A where A = { x, µ A (x, ν A (x /x R} is an ILFI. and Define A ( = { x, µ ( A Obviously, ν ( A (x, ν( A (x /x R/A} as follows: µ ( A (x + µ A = µ A (x ν ( A (x + ν A = ν A (x, for all x R. and µ( A are well-defined. Also A( is an ILFS(R/A. Theorem 3.5. Let A = { x, µ A (x, ν A (x /x R} be an ILFI. Then A ( is an ILFI of R/A, where A ( = { x, µ ( A (x, ν( A (x /x R/A} is defined by 23 Proof. Let x, y R. Then and µ ( A (x + µ A = µ A (x and ν ( A (x + ν A = ν A (x, for all x R. ν ( A ((x + ν A + (y + ν A = ν ( A (x + y + ν A = ν A (x + y ν A (x ν A (y ν ( A ((x + ν A(y + ν A = ν ( A (xy + ν A Hence A ( is an ILFI(R/A. = ν ( A (x + ν A ν ( A (y + ν A = ν A (xy ν A (x ν A (y = ν ( A (x + ν A ν ( A (y + ν A. Theorem 3.6. (Correspondence Theorem for ILFI Let f : R S be a ring epimorphism. Then there is a one-to-one order preserving correspondence between ILFI of S and the ILFI of R, which are constant on Ker f. Proof. Let F (R denote the set of ILFI of R which are constant on Ker f and F (S denote the set of ILFI of S. Define Φ : F (R F (S by Φ(A = f(a, for all A F (R and Ψ : F (S F (R by Ψ(B = f 1 (B for all B F (S. Global Journals Inc. (US

9 24 Then Φ and Ψ are well-defined and are inverses of each other, thus giving the one to one correspondence. It can easily be verified that this correspondence preserves the order too. Theorem 3.7. Let B = { x, µ B (x, ν B (x /x R} be an ILFSR and C be any ideal of R. Let be an ILFS of R/C, where D = { [x], µ D [x], ν D [x] /[x] R/C} µ D [x] = {µ B (z/z [x]}, ν D [x] = {ν B (z/z [x]}, for all x R, and [x] = x + C. Then D is an ILFSR of R/C. Proof. Let x, y R. Then Also ν D ([x] [y] = ν D ([x y] = {ν B (x y + z/z C} {ν B (x y + a b/a, b C} = {ν B ((x + a (y + b/a, b C} ( {ν B (x + a/a C} ( {ν B (y + b/b C} = ν D [x] ν D [y]. ν D ([x][y] = ν D ([xy] = {ν B (xy + z/z C} {ν B (xy + (xv + uy + uv/u, v C} = {ν B (x(y + v + u(y + v/u, v C} = {ν B ((y + v(x + u/u, v C} {(ν B (y + v (ν B (x + u/u, v C} = ( {ν B (x + u/u C} ( {ν B (y + v/v C} = ν D [x] ν D [y]. Hence D is an ILFSR of R/C. The ILFSR, D ={<[x], μμ D[x], ν[x]>/[x] belongs to R/C} is called the Intuitionistic L-fuzzy Quotient Subring of B relative to C and denoted as B/C and is abbreviated as ILFQSR. Global Journals Inc. (US

10 In this section we define an ILFI of an ILFSR and some elementary results are obtained. Also we discuss the ILFI of an ILFSR under an epimorphism. Definition 4.1. Let A = { x, µ A (x, ν A (x /x R} be an ILFS. Let B = { x, µ B (x, ν B (x /x R} be an ILFSR with A B. Then A is called an ILFI of B if for all x, y R, (i µ A (x y µ A (x µ A (y (ii µ A (xy (µ B (x µ A (y (µ A (x µ B (y (iii ν A (x y ν A (x ν A (y (iv ν A (xy (ν B (x ν A (y (ν A (x ν B (y 25 Since R is commutative, ν A (xy (ν B (x ν A (y (ν A (x ν B (y for all x, y R if and only if ν A (xy ν B (x ν A (y, for all x, y R. Definition 4.2. Let A = { x, µ A (x, ν A (x /x R} be an ILFI. Then A = {x R/µ A (x > 0, ν A (x = 0} is an ideal of R, if L is regular. The proof of the following result is direct. Theorem 4.3. Let B = { x, µ B (x, ν B (x /x R} be an ILFSR and A = { x, µ A (x, ν A (x /x R} be an ILFI of B. If L is regular, then A = {x R/µ A (x > 0, ν A (x = 0} is an ideal of B = {x R/µ B (x > 0, ν B (x = 0}. Theorem 4.4. Let A = { x, µ A (x, ν A (x /x R} be an ILFI of an ILFSR B = { x, µ B (x, ν B (x /x R}. Then A is an ILFSR. Proof. For x, y R For x, y R ν A (x y ν A (x ν A (y. ν A (xy (ν B (x ν A (y (ν A (x ν B (y (ν A (x ν A (y (ν A (x ν A (y = ν A (x ν A (y. Hence A is an ILFSR. Theorem 4.5. Let A = { x, µ A (x, ν A (x /x R} be an ILFI of R and B = { x, µ B (x, ν B (x /x R} be an ILFSR. Then A B is an ILFI of B. Global Journals Inc. (US

11 26 Proof. Clearly A B B and A B is an ILFSR of R. For x, y R, (ν A ν B (xy = ν A (xy ν B (xy [ν A (x ν A (y] [ν B (x ν B (y] = (ν A (x [ν B (x ν B (y] (ν A (y [ν B (x ν B (y] = ([ν A (x ν B (x] ν B (y ([ν A (y ν B (y] ν B (x ν B (x [ν A (y ν B (y] = ν B (x (ν A ν B (y. Therefore A B is an ILFI of B. Theorem 4.6. Let C = { x, µ C (x, ν C (x /x R} be an ILFSR and A = { x, µ A (x, ν A (x /x R}, B = { x, µ B (x, ν B (x /x R} be two ILFI of C. Then A B is an ILFI of C. Proof. Clearly A B C and A B is an ILFSR. For x, y R, (ν A ν B (xy = ν A (xy ν B (xy Therefore A B is an ILFI of C. (ν C (x ν A (y (ν C (x ν B (y = ν C (x (ν A (y ν B (y = ν C (x (ν A ν B (y. Theorem 4.7. Let B = { x, µ B (x, ν B (x /x R} be an ILFSR. Let A = { x, µ A (x, ν A (x /x R} be an ILFI of B. Let f : R S be an onto homomorphism. Then f(a is an ILFI of f(b. Proof. Clearly f(a and f(b are ILFSR of S and f(a f(b. Now for all x, y S, f(ν A (xy = {ν A (w : w R, f(w = xy} {ν A (uv : u, v R, f(u = x, f(v = y} {ν B (u ν A (v/f(u = x, f(v = y, u, v R} = ( {ν B (u/u R, f(u = x} ( {ν A (v/v R, f(v = y} = f(ν B (x f(ν A (y. Therefore f(a is an ILFI of f(b. Global Journals Inc. (US

12 Theorem 4.8. Let f : R S be an onto homomorphism. Let B = { x, µ B (x, ν B (x /x S} be an ILFSR of S and A = { x, µ A (x, ν A (x /x S} be an ILFI of B. Then f 1 (A is an ILFI of f 1 (B. Proof. Clearly f 1 (A and f 1 (B are ILFSR of R and f 1 (A f 1 (B. Now f 1 (ν A (xy = ν A (f(xy = ν A (f(xf(y ν B (f(x ν A (f(y = f 1 (ν B (x f 1 (ν A (y. 27 Therefore f 1 (A is an ILFI of f 1 (B. Here we define homomorphism and isomorphism of an ILFSR. The fundamental theorem of ring homomorphism and the Third Isomorphism Theorem for rings are established for ILFSR. Definition 5.1. Let A = { x, µ A (x, ν A (x /x R} be an ILFSR of R and B = { x, µ B (x, ν B (x /x S} be an ILFSR of S. (1 A weak homomorphism from A into B is an epimorphism f of R onto S such that f(a B. If f is a weak homomorphism of A into B, then A is said to be weakly homomorphic to B and written as A f B or A B. (2 A weak isomorphism from A into B is a weak homomorphism f from A into B which is also an isomorphism of R onto S. If f is a weak isomorphism from A into B, then A is said to be weakly isomorphic to B and written as A f B or A B. (3 A homomorphism from A onto B is a weak homomorphism f from A onto B such that f(a = B. If f is a homomorphism of A onto B, then A is said to be homomorphic to B and written as A f B or A B. (4 An isomorphism from A onto B is a weak isomorphism f from A into B such that f(a = B. If f is an isomorphism from A onto B, then A is said to be isomorphic to B and written as A f = B or A =B. Let B = { x, µ B (x, ν B (x /x R} be an ILFSR of R. Let A = { x, µ A (x, ν A (x /x R} be an ILFI of B. Assume that L is regular. Then it is clear that A is an ideal of B and B/B is an ILFSR of B. Thus we can consider the ILFQSR of B/B relative to A. This ILFQSR is denoted as B/A. Global Journals Inc. (US

13 28 Theorem 5.2. Let B = { x, µ B (x, ν B (x /x R} be an ILFSR and A = { x, µ A (x, ν A (x /x R} be an ILFI of B. Suppose that L is regular. Then B/B B/A. Proof. Let f be the natural homomorphism from B onto B /A. Then f(ν B /ν B ([y] = {(ν B /ν B (x/x B, f(x = [y]} for all y B where [y] = y + A. Therefore B/B f B/A. = {ν B (z/z [y]} = (ν B /ν A ([y] The result of the following theorems are proved for membership and non membership functions. Theorem 5.3. Let B = { x, µ B (x, ν B (x /x R} and C = { x, µ C (x, ν C (x /x S} be an ILFSR of the rings R and S such that B C. Suppose that L is regular. Then there exists an ILFI A = { x, µ A (x, ν A (x /x R} of B such that B/A = C/C. Proof. Since B C, there exists an epimorphism f of R onto S such that f(b = C. Define an ILFS, A = { x, µ A (x, ν A (x /x R} as follows: and µ B (x, x Ker f µ A (x = 0, otherwise, for all x R 0 x Ker f ν A (x = ν B (x, otherwise, for all x R Clearly A is an ILFSR and A B. If x Kerf then µ A (xy = µ B (xy µ B (x µ B (y µ B (x µ A (y for all y R. If x R\ Kerf, then µ A (x = 0. Hence µ A (xy µ B (x µ A (y Global Journals Inc. (US

14 for all y R. If x Kerf, then ν A (x = 0 and so for all y R. If x R\ Kerf, then ν A (xy ν B (x ν A (y ν A (xy = ν B (xy ν B (x ν B (y ν B (x ν A (y for all y R. Hence A is an ILFI of B. Since B C, f(b = C. Let g = f/b. Then g is a homomorphism of B onto C and Kerg = A. Thus there exists an isomorphism h of B /A onto C such that h([x] = g(x for all x B. For such an h, we have and Therefore B/A h = C/C. h(µ B /µ A (y = {(µ B /µ A [x]/h([x] = y, x B } = { {µ B (z/z [x]}/g(x = y, x B } = {µ B (z/z B, g(z = y} = {µ B (z/z R, f(z = y} = f(µ B (y = µ C (y, for all y C. h(ν B /ν A (y = {(ν B /ν A [x]/h([x] = y, x B } = { {ν B (z/z [x]}/g(x = y, x B } = {ν B (z/z B, g(z = y} = {ν B (z/z R, f(z = y} = f(ν B (y = ν C (y, for all y C. 29 Theorem 5.4. Let A = { x, µ A (x, ν A (x /x R}, B = { x, µ B (x, ν B (x /x R} and C = { x, µ C (x, ν C (x /x R} be ILFSR. Let A be an ILFI of B and A, B be ILFI of C. Suppose that L is regular. Then (C/A/(B/A = C/B. Global Journals Inc. (US

15 Proof. Clearly A is an ideal of B and A, B are ideals of C. By the Third Isomorphism Theorem for Rings, where f is given by (C /A /(B /A f = C /B, f(x + A + (B /A = x + B for all x C. Thus 30 f((µ C /µ A /(µ B /µ A (x + B = ((µ C /µ A /(µ B /µ A (x + A + (B /A = {(µ C /µ A (y + A /y C, y + A x + A + (B /A } = { {µ C (z/z y + A }/y C, y + A x + A + (B /A } = {(µ C (z/z C, z + A x + A + (B /A } = {(µ C (z/z x + A + (B /A } = {(µ C (z/z C, f(z x + B } = (µ C /µ B (x + B for all x C. and f((ν C /ν A /(ν B /ν A (x + B = ((ν C /ν A /(ν B /ν A (x + A + (B /A = {(ν C /ν A (y + A /y C, y + A x + A + (B /A } = { {ν C (z/z y + A }/y C, y + A x + A + (B /A } = {(ν C (z/z C, z + A x + A + (B /A } = {(ν C (z/z x + A + (B /A } = {(ν C (z/z C, f(z x + B } = (ν C /ν B (x + B for all x C. Hence (C/A/(B/A f = C/B. [1] Adam Niewiadomski and Eulalia Szmidt, Handling Uncertainty in natural sentences via IFS, on Intuitionistic Fuzzy Sets, 7 (2001, No.1, [2] Afshar Alam, Sadia Hussian and Yasir Ahmad, Applying Intuitionistic Fuzzy Approach to Reduce Search Domain in a Accidental Case, IJACSA, 1, No.4 (2010. Global Journals Inc. (US

16 [3] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1 (1986, [4] K. T. Atanassov, New Operations defined over the Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 61 (1994, [5] K. T. Atanassov, Fuzzy sets, Theory and Applications, Studies in fuzziness and soft computing, 35, Physica-Verlag, Heidelberg, [6] D. Boyadzhieva, B. Kolev and N. Netov, Intuitionistic fuzzy data warehouse and some analytical operations, IS, 5 th IEEE International Conference (2010, [7] D. Ćoker, An Introduction to Intuitionistic fuzzy topological spaces, FSS, 88 (1997, [8] Eulalia Szmidt and Januz Kaprzyk, Inter-Fuzzy set in some Medical Applications, NIFS, 7(2001, 4, [9] Y. B. Jun and K. H. Kim, Intuitionistic fuzzy ideals of semigroups, Indian Journal of Pure Applied Math., 33 (2002 4, [10] Magdaléna Renčová, An example of Applications of IFS to sociometry, Bulgarian Academy of Sciences, Cybernetics and Technologies, Vol 9, No.2 (2009. [11] D. S. Malik and J. N. Mordeson, Extensions of fuzzy subrings and fuzzy ideals, FSS, 45 (1992, [12] D. S. Malik and J. N. Mordeson, Fuzzy Commutative Algebra, 1998, World Scientific. [13] K. Meena and K. V. Thomas, Intuitionistic L-fuzzy Subrings, IMF, 6 (2011, No.52, [14] A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl., 35 (1971, [15] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965, Global Journals Inc. (US