Global Journal of Pure and Applied . ISSN 0973-1768 Volume 13, Number 4 (2017), pp. 1303–1317 © Research India Publications http://www.ripublication.com/gjpam.htm

Correspondence and Theorems for L-Intuitionistic or L-Vague Sub Rings

G. Vasanti Basic Science and Humanities Department, Aditya Institute of Technology And Management (An Autonomous Institution), Tekkali, Andhra Pradesh, 532201.

Abstract The aim of this paper is basically to generalize such results as First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, and Corre- spondence Theorem for the L-intuitionistic fuzzy or L-vague fuzzy sub rings of a under a crisp map.

AMS subject classification: 03F55, 06C05, 16D25. Keywords: Intutionistic L-fuzzy or L-vague Sub ring, Correspondence Theorem, First (Second, Third) Isomorphism Theorem.

1. Introduction Zadeh [1], in his pioneering paper, introduced the notion of Fuzzy Subset of a set X as a function µ from X to the closed interval [0,1] of real numbers. The function µ,he called, the membership function which assigns to each member x of X its membership value, µx in [0, 1]. In 1983, Atanassov [2] generalized the notion of Zadeh fuzzy subset of a set further by introducing an additional function ν which he called a non membership function with some natural conditions on µ and ν, calling these new generalized fuzzy subsets of a set, intutionistic fuzzy subsets. Thus according to him an intutionistic fuzzy subset of a set X, = is a pair A (µA,νA), where µA,νA are functions from the set X to the closed interval [ ] ∈ + ≤ 0, 1 of real numbers such that for each x X, µx νx 1, where µA is called the membership function of A and νA is called the non membership function of A. Later on in 1984, Atanassov and Stoeva [3], further generalized the notion of intuitionistic fuzzy subset to L-intuitionistic fuzzy subset, where L is any complete lattice with a complete 1304 G. Vasanti order reversing involution N. Thus an L-intutionistic fuzzy subset A of a set X, is a pair → ≤ (µA,νA) where µA,νA: X L are such that µA NνA. Let us recall that a complete order reversing involution is a map N: L → L such that (1) N0L =1L and N1L =0L (2) α ≤ β implies Nβ ≤ Nα (3) NNα = α (4) N(∨i∈I αi) = ∧i∈I Nαi and N(∧i∈I αi) = ∨i∈I Nαi. Liu [4] in 1982 introduced the concept of fuzzy ring and fuzzy . In 1985, Ren [5] examined the concepts of fuzzy ideal and fuzzy ring, which were actually an extension of Rosenfeld’s [6] fuzzy by starting with an ordinary ring and then define a fuzzy sub ring based on the ordinary operations of the given ring. Coming to generalizations of substructures of rings on to the if/v-subsets, we briefly mention the following papers: Banergee-Basnet [7] in 2003, introduced the concepts of if/v-sub rings and if/v-ideals of a ring. Hur-Kang-Song [8] introduced the concepts of if/v-sub ring of a ring, if/v- ideals and study of if/v-prime/maximal ideals, if/v-nil radicals etc. were made in Hur et al. [9], Jun-Ozturk-Park [10] and Wang-Lin [11] etc.. Palaniappan, Arjunan and Palanivelrajan [12] in 2008 studied the concepts of ho- momorphism and anti- in intuitionistic L-fuzzy sub rings. In the same year, Meena and Thomas [13] presented properties of L-if/v-sub rings, L-if/v-ideals and introduced the . Murthy-Vasanti [14] presented a detailed study of the (lattice) algebraic properties of L-if/v-images and L-if/v-inverse images under a crisp map for L-if/v- that take their truth values in a complete lattice and Vasanti [15] gave an exclusive study of the (lattice) algebraic properties of L-if/v-images and L-if/v-inverse images under a crisp map for L-if/v-sub rings that take their truth values in a complete lattice. In this paper, we apply parts of the above Theory to generalize such results as First Isomorphism Theorem, Second Isomorphism Theorem, Third Isomorphism theorem, Correspondence Theorem etc.. Hence, the set of all L-if/v-subsets of a set X be denoted by AL(X). For any pair = = ≤ ≤ of L-if/v-subsets A (µA,νA) and B (µB,νB) of X, A B iff µA µB and νB ≤ νA, AL(X) becomes a complete infinitely distributive lattice, provided L is also a complete infinitely distributive lattice. For any family (Ai)i∈I of L-if-subsets of X, =∨∈ ∨ =∧∈ ∨ ∈ = ∨ µ∨i∈I Ai i I µAi and ν i∈I Ai i I νAi where i I Ai (µ∨i∈I Ai ,ν i∈I Ai ) and ∧ ∈ ∧ =∨∈ ∧ ∈ = ∧ µ∧i∈I Ai = i I µAi and ν i∈I Ai i I νAi where i I Ai (µ∧i∈I Ai ,ν i∈I Ai ). Let us recall that (a) a complete lattice L is a complete infinite distributive lattice iff ∧ ∨ =∨ ∧ for all subsets (αi)i∈I , (βj )j∈J and elements α, β of L (1) α ( j∈J βj ) j∈J (α βj ), (2) β ∨ (∧i∈I αi) =∧i∈I (β ∨ αi), hold. Consequently, in a complete infinite distributive lattice, for all subsets (αi)i∈I , ∨ ∨ ∧ = ∨ ∧ ∨ ∧ ∧ ∨ (βj )j∈J , (3) i∈I j∈J (αi βj ) ( i∈I αi) ( j∈J βj ) and (4) i∈I j∈J (αi βj ) ∧ ∨ ∧ = ( i∈I αi) ( j∈J βj ) hold and (b) a complete lattice with a unary complement op- eration is a complete de Morgan lattice iff for all subsets (αi)i∈I and (βj )j∈J of L, (1) ∨ c =∧ c ∧ c =∨ c ( i∈I αi) i∈I αi and (2) ( j∈J βj ) j∈J βj hold. = For any set X, the L-if-subset (µX,νX) (1X, 0X), where 1X or simply 1 is the constant map assuming the value 1 of L for each x ∈ X and 0X or simply 0 is the constant Correspondence And Isomorphism Theorems For L-Intuitionistic 1305 map assuming the value 0 of L for each x ∈ X, which turns out to be the largest element in AL(X). The L-if-empty subset φ of X is given by (µφ,νφ) = (0, 1), which is the least element in AL(X). Also for any µ: X → L, both (µ, Nµ), where Nµ: X → L is defined by (Nµ)(x) = N(µx) ∀x ∈ X and (Nµ, µ), define L-if-subsets of X because = N is an involution on L.ForA (µA,νA) an L-if-subset of X, since N is an order reversing involution, (νA,µA) is also an L-if-subset of X called as the L-if-complement of A, denoted by Ac. Observe that Ac = X − A = X ∧ Ac. Further for any pair A, B of L-if/v-subsets of X, we define B/A to be B ∧ Ac.

2. Basic Results In this section, first we state some definitions and statements from [15], which are useful in the main results. Definition 2.1. (a) An L-if/v-subset A of R is called an L-if/v-sub ring of R iff: ≥ ∧ ≤ ∨ ∈ (1) µA(xy) µA(x) µA(y) and νA(xy) νA(x) νA(y) for each x,y R. + ≥ ∧ + ≤ ∨ (2) µA(x y) µA(x) µA(y) and νA(x y) νA(x) νA(y) for each x,y ∈ R. − ≥ − ≤ ∈ (3) µA( x) µA(x) and νA( x) νA(x) for each x R. { ∈ = = (b) For any L-if/v-sub ring A of a ring R, A∗ = x R/µA(x) µA(0) and νA(x) } ∗ { ∈ } νA(0) and A = x R/µA(x) > 0 and νA(x) < 1 . (c) A complete lattice L is strongly regular iff0is∧ - prime or α>0, β>0 implies α ∧ β>0and1is∨- prime or α<1, β<1 implies α ∨ β<1. : → (d) Let X, Y be a pair of sets and let f X Y be a map. Let A =(µA, νA) and B =(µB,νB) be L-if/v-subsets of X and Y respectively. → ∨ −1 ∧ −1 (1) Let µD,νD : Y L be defined by µDy = µAf y and νDy = νAf y ∀y ∈ Y . Then D is a well defined L-if/v-subset of Y , called as the L- intutionistic fuzzy/vague of A under f or simply the L-if/v-image of A under f . → ∀ ∈ (2) Let µC,νC : Y L be defined by µCx = µBfxand νCx = νBfx, x X. Then C is a well defined L-if/v-subset of X, called as the L-intutionistic fuzzy/vague inverse image of B under f or simply the L-if/v-inverse image of B under f . ∀ ∈ =∨ −1 = = = (e) Clearly, y Y , µfXy 1Xf y χ fXy or µfX χ fX and νfXy ∧ −1 = = = 0Xf y NχfXy or νfX NχfX implying fX (χ fX,NχfX). (f) For any L-if/v-sub ring A of a ring R, 1306 G. Vasanti

≥ ≤ (1) A is an L-if/v-left-ideal of R iff µA(xy) µAy and νA(xy) νA(y) for each x,y ∈ R. ≥ ≤ (2) A is an L-if/v-right-ideal of R iff µA(xy) µAx and νA(xy) νA(x) for each x,y ∈ R. (3) A is an L-if/v-ideal of R iff the A is an L-if/v-right-ideal and L-if/v-left-ideal ≥ ≥ ≥ ∨ of R or ( µA(xy) µAx and µA(xy) µAy)or(µA(xy) µAx µAy). (g) A is an L-if/v-sub ring of R implies ≥ ≤ ∈ (1) µA(0) µA(x) and (2) νA(0) νA(x) for each x R. (h) A is an L-if/v-sub ring (left ideal, right ideal, ideal) of R implies ={ ∈ = = } (1) A∗ x R/µA(x) µA(0), νA(x) νA(0) is a sub ring (left ideal, right ideal, ideal) of R; ∗ ={ ∈ } (2) A x R/µA(x) > 0, νA(x) < 1 is a sub ring (left ideal, right ideal, ideal) of R whenever L is strongly regular. Lemma 2.2. For any pair of rings R and S and for any crisp homomorphism f : R → S the following are true: 1. A is an L-if/v-sub ring (left ideal, right ideal, ideal) of R implies f (A) is an L- if/v-sub ring (left ideal, right ideal, ideal) of S, whenever L is a complete infinite distributive lattice (and f is onto) − 2. B is an L-if/v-sub ring (left ideal, right ideal, ideal) of S implies f 1(B) is an L-if/v-sub ring (left ideal, right ideal, ideal) of R.

Lemma 2.3. For any family of L-if/v-sub rings (left ideals, right ideals, ideals) (Ai)i∈I of a ring R,

(1) ∧i∈I Ai is an L-if/v-sub ring (left ideals, right ideals, ideals) of R.

(2) ∨i∈I Ai is an L-if/v-sub ring (left ideal, right ideal, ideal) of R whenever (Ai)i∈I is a sup/inf assuming chain.

Theorem 2.4. For any L-if/v-sub ring B of a ring R and for any crisp ideal N of R, the → ∈ + =∨ + L-if/v-subset C: R/N L where for each x R, µC(x N) µB(x N) and νC(x + N) =∧νB(x + N),isanL-if/v-sub ring of R/N when L is a complete infinite distributive lattice.

Definition 2.5. For any L-if/v-sub ring B of a ring R and for any ideal N of R, the L- if/v-sub ring C : R/N → L, where L is a complete infinite distributive lattice, defined + =∨ + + =∧ + ∈ by µC(x N) µB(x N)and νC(x N) νB(x N)for each x R, is called B the L-if/v-quotient sub ring of R/N relative to B and is denoted by B/N or . N Correspondence And Isomorphism Theorems For L-Intuitionistic 1307

In other words when N is an ideal of R and B is any L-if/v-sub ring of R, and B R L is a complete infinite distributive lattice, : → L is defined by µ B (r + N) = N N N ∨ + + =∧ + ∈ µB(r N) and ν B (r N) νB(r N) for each r R. N For any pair of L-if/v-sub rings A and B of a ring R such that A ≤ B, ∈ ≥ ∧ (i) A is called an L-if/v-left ideal of B iff for each x,r R, µA(rx) µB(r) ≤ ∨ µA(x) and νA(rx) νB(r) νA(x). ∈ ≥ ∧ (ii) A is called an L-if/v-right ideal of B iff for each x,r R, µA(xr) µB(r) ≤ ∨ µA(x) and νA(xr) νB(r) νA(x). (iii) A is called an L-if/v-ideal of B iff A is both L-if/v-left ideal and L-if/v-right ideal of B.

Theorem 2.6. For any pair of L-if/v-sub rings A and B of a ring R such that A is an L-if/v-(left, right)ideal of B:

1. A∗ is an (left, right) ideal of B∗. ∗ ∗ 2. A is an (left, right) ideal of B whenever L is strongly regular.

Lemma 2.7. For any pair of L-if/v-sub rings A and B of a ring R such that A is ∗ B ∗ an L-if/v ideal of B, the L-if/v-subset C: → L defined by, for each b ∈ B A∗ + ∗ =∨ + ∗ + ∗ =∧ + ∗ µC(b A ) µB(b A ) and νC(b A ) νB(b A ),isanL-if/v-sub ring of B∗ , whenever L is a strongly regular complete infinite distributive lattice. A∗ Definition 2.8. For any pair of L-if/v-sub rings A and B of a ring R such that A is an L-if/v ideal of B and L is a strongly regular complete infinite distributive lattice, the L- ∗ ∗ B if/v-quotient sub ring of B|B relative to A , denoted by B/A or , is defined by B/A: ∗ ∗ ∗ ∗ A ∗ ∗ B /A → L with µ (b + A ) =∨µ (b + A ) and νB/A(b + A ) =∧νB(b + A ) ∗ B/A B for each b ∈ B and is called L-if/v-quotient sub ring of B relative to A.

Lemma 2.9. For any pair of L-if/v-sub rings A and B of a ring R such that A is an ∗ ∗ ∗ L-if/v-ideal of B, (B/A) = B /A .

Theorem 2.10. For any L-if/v-(left, right) ideal A of R and an L-if/v-sub ring B of R, A ∧ B is an L-if/v-(left, right) ideal of B.

3. Main Results In this section, we begin with a result on L-if/v-(inverse) image of an L-if/v-ideal. Then we go on to introduce various notions of L-if/v-homo (iso) morphisms. Finally, in the end 1308 G. Vasanti

∗ ∗ we show that ηA = B implies ηA = B , under certain conditions. Then we generalize the First Isomorphism Theorem, Correspondence Theorem, the Second Isomorphism Theorem and the Third Isomorphism Theorem in that order.

Theorem 3.1. For any pair of rings R1 and R2 and for any pairs of L-if/v-sub rings A, B of R1 and C, D of R2, for any η: R1 → R2 such that A is an L-if/v-ideal of B and C is a L-if/v-ideal of D,

1. ηA is an L-if/v-ideal of ηB whenever L is a complete infinite distributive lattice − − 2. η 1C is an L-if/v-ideal of η 1D.

Proof.

(1) Since A, B are L-if/v- of R1, by 2.2(1), ηA, ηB are L-if/v-subrings of R2. Further, A ≤ B implies ηA ≤ ηB.SoηA is an L-if/v- of ηB. Since (a) L is a complete infinite distributive lattice, (b) ηu = x, ηv = y impliy − uv ∈ η 1(xy) and (c) A is an L-if/v-ideal of B, we get that ∨ ≥∨ ≥∨ ∧ µηA(xy) = z∈η−1(xy)µAz u∈η−1x,v∈η−1yµA(uv) u∈η−1x,v∈η−1y(µB(u) ∨ ∧ ∨ ∧ µA(v)) = ( v∈η−1xµB(v)) ( u∈η−1yµA(v)) = µηB(x) µηA(y) and ∧ ≤∧ ≤∧ ∨ νηA(xy) = z∈η−1(xy)νAz u∈η−1x,v∈η−1yνA(uv) u∈η−1x,v∈η−1y(νB(u) ∧ ∨ ∧ ∨ νA(v)) = ( u∈η−1xνB(u)) ( v∈η−1yνA(v)) = νηB(x) νηA(y). Hence µηA(xy) ≥ ∧ ≤ ∨ ≥ µηB(x) µηA(y) and νηA(xy) νηB(x) νηA(y). And similarly µηA(yx) ∧ ≤ ∨ µηB(x) µηA(y) and νηA(yx) νηB(x) νηA(y).Hence ηA is an L-if/v-ideal of ηB whenever L is a complete infinite-distributive lattice.

−1 −1 (2) Since C, D are L-if/v-subrings of R2, by 2.2(2), η C, η D are L-if/v-subrings −1 −1 −1 of R1. Further, C ≤ D implies η C ≤ η D. So, η C is an L-if/v-subring of −1 η D. Since C is an L-if/v-ideal of D, for each x,y ∈ R1, ≥ ∧ ∧ µη−1C(xy) = µCη(xy) = µC(η(x)η(y)) µD(ηx) µC(ηy) = µη−1D(x) ≤ ∨ µη−1C(y) and νη−1C(xy) = νCη(xy) = νC(η(x)η(y)) νD(ηx) νC(ηy) = ∨ ≥ ∧ νη−1D(x) νη−1C(y). Hence µη−1A(xy) µη−1D(x) µη−1C(y) and νη−1C(xy) ≤ ∨ ≥ ∧ νη−1D(x) νη−1C(y). And similarly µη−1C(yx) µη−1D(x) µη−1C(y) and ≤ ∨ −1 −1  νη−1C(yx) νη−1D(x) νη−1C(y). Hence η C is an L-if/v-ideal of η D.

Definition 3.2. Let R1 and R2 be rings and let A be an L-if/v-sub ring of R1 and B be an L-if/v-sub ring of R2.

(a) A homomorphism η: R1 → R2 is called a L-if/v-weak homomorphism of A into B iff ηA ≤ B.Ifη is a L-if/v-weak homomorphism of A into B, then we say that A is L-if/v-weak homomorphic to B.

(b) An isomorphism η: R1 → R2 is called an L-if/v-weak isomorphism of A into B iff ηA ≤ B.Ifη is an L-if/v-weak isomorphism of A into B, then we say that A is L-if/v-weak isomorphic to B. Correspondence And Isomorphism Theorems For L-Intuitionistic 1309

(c) An epimorphism η: R1 → R2 is called an L-if/v-epimorphism of A onto B iff ηA = B.Ifη is an L-if/v-epimorphism of A onto B, then we say that A is L-if/v- epimorphic to B.

(d) An isomorphism η: R1 → R2 is called an L-if/v-isomorphism of A onto B iff ηA = B.Ifη is an L-if/v-isomorphism of A onto B, then we say that A is L-if/v- isomorphic to B. In Isomorphism Theorems we need some kind of one-one ness of the CF-operator N at the lower extreme 0L of L in the sense that α>0 implies Nα <1. Note that, since 1L is the largest element of L and 0L is the smallest element of L, the former statement is equivalent to Nα = 1(=N0) implies α = 0, as shown in the following:

Since α ≥ 0L, α = 0 implies α>0 which implies Nα <1, which is a contradiction. On the other hand, since Nα ≤ 1L, Nα <1 implies Nα = 1 but then α = 0, which is a contradiction.

(e) α>0 implies Nα < 1 if and only if Nα = 1 implies α =0.

(f) For any complete lattice L with the CF operator N, N is one-one at 0L for L if and only if Nα(= N(0)) = 1 implies α =0.

Theorem 3.3. For any pair of L-if/v-sub rings A and B of a ring R such that A is ∗ an L-if/v-ideal of B, B|B is homomorphic to B/A, whenever L is a strongly regular complete infinite distributive lattice. ∗ ∗ ∗ ∗ ∗ Proof. By 2.6, A is an L-if/v-ideal of B implies A is an ideal of B . Let η:B →B /A ∗ be the natural homomorphism which is onto. we will show that η(B|B ) = B/A: ∗ ∗ B /A → L.

∗ −1 ∗ ∗ ∗ (1) µ | ∗ (b + A ) = ∨µ | ∗ η (b + A ) = ∨µ (b + A ) = µ (b + A ) for each η(B B∗ ) B B B B/A b ∈ B where the first equality is due to the definition of L-if/v-image, the second − ∗ ∗ ∗ ∗ ∗ equality is due to η 1(b + A ) = {b ∈ B /ηb = b + A } = {b ∈ B /b + A = ∗ ∗ b + A } = b + A and the last equality is due to the definition 2.7 of B/A. Hence µη(B|B∗) = µB/A.

∗ −1 ∗ ∗ ∗ (2) νη(B|B∗)(b + A ) = ∧νB|B∗ η (b + A ) = ∧νB(b + A ) = νB/A(b + A ) for each ∗ b ∈ B or νη(B|B∗) = νB/A. ∗ ∗ So, η(B|B ) = B/A. Thus η: B|B → B/A is an onto homomorphism. 

Lemma 3.4. For any epimorphism η: R1 → R2 and for any pair of L-if/v-subsets A of ∗ ∗ R1 and B of R2 such that η(A) = B, η(A ) = B if the CF-operator N is one-one at 0L for L.

−1 −1 Proof. Let η(A) = B. Then µ (y) = ∨µ η y and νη(A)(y) = ∧νAη y.Nowwe ∗ ∗ η(A) A show that η(A ) = B . 1310 G. Vasanti

∈ ∗ ∈ ∗ ∨ −1 ≥ α η(A ) implies α = ηa, a A .NowµηAα = µηAηa = µAη ηa µAa>0 and −1 ∗ ∗ ∗ νηAα = νηAηa = ∧νAη ηa ≤ νAa<1. Therefore α ∈ B . Hence η(A ) ⊆ B . ∈ ∗ ∨ −1 Suppose α B . Then µBα>0 which implies µη(A)α>0 implying µAη α −1 = ∈ −1 ∨ −1 > 0. Since η is onto η α φ. If for each x η α, µAx = 0, then µAη α =0 ∈ −1 which is a contradiction. Therefore, there exists x0 η α such that µAx0 > 0. Now, 0 <µ x0 implies Nµ x0 < 1 because N is one-one at 0L for L. But νAx0 ≤ Nµ x0. A A ∗ ∗ ∗ ∗ A So, νAx0 < 1. Therefore x0 ∈ A and α = ηx0 ∈ ηA implying B ⊆ η(A ).Therefore ∗ ∗ η(A ) = B . 

4. First Isomorphism Theorem

Theorem 4.1. For any L-if/v-sub ring B of a ring R1 and for any L-if/v-sub ring C of a ring R2 such that B is L-if/v-epimorphic to C, there exists an L-if/v-ideal A of B ∗ such that B/A is L-if/v-isomorphic to C|C , whenever L is a strongly regular infinite distributive lattice with CF operator N which is one-one at 0L for L. Proof. Since (1) B is L-if/v-epimorphic to C (2) L is a strongly regular infinite distribu- tive lattice with CF-operator N which is one-one at 0L for L and (3) 3.2(c) and 3.4, there ∗ ∗ exists an epimorphism η: R1 → R2 such that η(B) = C and η(B ) = C . : → = ∈ Define A R1 L by µAx µBx, νAx = νBx when x Kerη and µAx =0L, νAx =1L when x ∈ Kerη. First we show that A is an L-if/v-subring of R1 or to show that ≥ ∧ ≤ ∨ − ≥ ∧ (i) µA(xy) µAx µAy, νA(xy) νAx νAy, (ii) µA(x y) µAx µAy, νA(x − y) ≤ νAx ∨νAy.

Without loss of generality both x, y be in Kerη. Then µAx = µBx, νAx = νBx and µAy − ∈ = µBy, νAy = νBy and xy, x y Kerη. Thus ≥ ∧ ∧ ≤ ∨ (i) µA(xy) = µB(xy) µBx µBy = µAx µAy and νA(xy) = νB(xy) νBx νBy = νAx ∨νAy − − ≥ ∧ ∧ − − (ii) µA(x y) = µB(x y) µBx µBy = µAx µAy and νA(x y) = νB(x y) ≤ νBx ∨νBy = νAx ∨νAy. Now we show that A is an L-if/v-ideal of B or to show that ≥ ∧ ≤ ∨ (i) µA(xy) µBx µAy and νA(xy) νBx νAy ≥ ∧ ≤ ∨ (ii) µA(yx) µBx µAy and νA(yx) νBx νAy.

Once again without loss of generality both x, y be in Kerη. Then µAx = µBx, νAx = ∈ νBx and µAy = µBy, νAy = νBy. Then xy Kerη. Thus ≥ ∧ ∧ ≤ ∨ (i) µA(xy) = µB(xy) µBx µBy = µBx µAy and νA(xy) = νB(xy) νBx νBy = νBx ∨νAy Correspondence And Isomorphism Theorems For L-Intuitionistic 1311

≥ ∧ ∧ ≤ (ii) µA(yx) = µB(yx) µBy µBx = µBx µAy and νA(yx) = νB(yx) νBy ∨νBx = νBx ∨νAy. Hence A is an L-if/v-ideal of B. ∗ ∗ ∗ ∗ ∗ Since B is L-if/v-epimorphic to C, ηB = C and ηB = C . Let φ = η|B . Then φ:B →C ∗ is an epimorphism and Kerφ = A as follows: ∗ g ∈ Kerφ implies g ∈ B and ηg = φg = 0 implies g ∈ Kerη or µ g = µ g and νAg = ∗ A ∗ B νBg. Since g ∈ B , µ g = µ g>0L and νAg = νBg<1L. So, g ∈ A . ∗ A∗ B ∗ ∗ Conversely, A ⊆ B , φ = η|B and the definition of A imply if g ∈ A then µ g> A ∗ 0 which implies g ∈ Kerη. But ηg = 0 implies φg =0org ∈ Kerφ. Thus Kerφ = A . L ∗ ∗ B B ∗ Thus by (crisp) First IsomorphismTheorem, there is an isomorphism h: = →C A∗ Kerφ ∗ such that h(b + A ) = φb. B ∗ − ∗ Now we show that h( ) = C. But first we show that for all z ∈ C ,ifh 1z = b +A A 0 ∗ −1 ∗ or z = h(b0 + A ) then η z = b0 + A , as follows: −1 ∗ ∗ ∗ b ∈ η z implies ηb = z which implies h(b + A ) = φb = ηb = z = h(b0 + A ) or b + A ∗ ∗ ∗ −1 ∗ = b0 + A or b ∈ b0 + A because 0 ∈ A . Thus η z ⊆ b0 + A . ∗ ∗ ∗ Conversely, b0 +a ∈ b0 +A , a ∈ A implies η(b0 +a) = φ(b0 +a) = h(b0 +a +A ) ∗ ∗ −1 ∗ −1 = h(b0 + A ) = z. Thus for a ∈ A , b0 + a ∈ η z or b0 + A ⊆ η z. Now, since L is a strongly regular complete infinite distributive lattice, the definitions of the L-if/v-image and the L-if/v-quotient B/A imply: ∨ −1 + ∗ ∨ + ∗ µ B (z) = µ B (h z) = µ B (b0 A ) = µB(b0 A ) and h( A ) A A −1 ∗ ∗ ν B (z) = ∧ν B (h z) = ν B (b0 + A ) = ∧νB(b0 + A ). h( A ) A A On the other hand, the definition of the L-if/v-image, ηB implies: ∨ −1 ∧ −1 µCz = µηBz = µB(η z) and νCz = νηBz = νB(η z). −1 + ∗ −1 + ∗ Now, since h z = b0 A implies η z = b0 A , µ B (z) = µCz and ν B (z) = νCz   h( A ) h( A ) B B ∗ or h = C or is L-if/v-isomorphic to C|C .  A A Lemma 4.2. For any L-if/v-sub ring A of a ring R, − ⇒ − ⇒ (1) µA(x1 x2) = µA0 µA(x1) = µA(x2). (2) νA(x1 x2) = νA0 νA(x1) = νA(x2). Proof. By 2.1(a) and 2.1(g), = − + ≥ − ∧ ∧ (1) µA(x1) µA(x1 x2 x2) µA(x1 x2) µA(x2) = µA0 µAx2 = µAx2 − + because µA0 is the largest of µAR. On the other hand, µA(x2) = µA(x2 x1 x1) ≥ − ∧ − − ∧ − ∧ µA(x2 x1) µA(x1) = µA( (x1 x2)) µA(x1) = µA(x1 x2) µA(x1) ∧ = µA0 µAx1 = µAx1. Therefore µA(x1) = µA(x2).

(2) νA(x1) = νA(x1 − x2 + x2) ≤ νA(x1 − x2) ∨ νA(x2) = νA0 ∨ νAx2 = νAx2 because νA0 is the least of νAR. On the other hand, νA(x2) = νA(x2 − x1 + x1) ≤ νA(x2 − x1) ∨ νA(x1) = νA(−(x1 − x2)) ∨ νA(x1) = νA(x1 − x2) ∨ νA(x1) = νA0 ∨ νAx1 = νAx1. Therefore νA(x1) = νA(x2).  1312 G. Vasanti

5. Correspondence Theorem

Theorem 5.1. For any pair of rings R1 and R2 and for any epimorphism η:R1→R2, → A ηA defines a one-one correspondence between, the set ζ K (R1) of all L-if/v-sub rings of R1 that are constant on the Ker(η) and the set ζ(R2) of all L-if/v-subrings of R2 in such a way that: ≤ ≤ ∈ (a) A1 A2 iff ηA1 ηA2 for all A1,A2 ζ K (R1).

(b) A is an L-if/v-ideal of R1 iff ηA is an L-if/v-ideal of R2.

Proof.

−1 (1) Since η is surjective, for each B ∈ R2, ηη B = B.

(2) If A is an L-if/v-subring of R1 and A is constant on Ker(η) then for each x ∈ ∈ Ker(η), µAx = µA0 and νAx = νA0 because 0 Ker(η). ∈ −1 ∈ −1 (3) Next for any y R2, µA is constant on η y because, x1,x2 η y implies ηx1 − ∈ − = ηx2 which implies x1 x2 Ker(η) or µA(x1 x2) = µA0orµA(x1) = µA(x2), by the previous Lemma. −1 −1 Thus for any y ∈ R2, ∨µ η y = µ x. Similarly ∧νAη y = νAx where x ∈ − A A η 1y is any element. − (4) Now we show that η 1ηA = A. −1 ∨ −1 ∧ −1 ∈ Let B = ηA and C = η B. Then µBy = µAη y, νBy = νAη y, for each y R2. ∨ −1 ∧ −1 And µCx = µBηx = µAη ηx = µAx, νCx = νBηx = νAη ηx = νAx, since A −1 is constant on η y, for each y ∈ R2. − Thus η 1ηA = A when A is constant on ker(η). From (1) and (4) we shall get that A → ηA is a one-one correspondence between the set ζ K (R1) of all L-if/v-subrings of R1 that are constant on the Ker(η) and the set ζ(R2) of all L-if/v-subrings of R2. ∈ (a) Let A1,A2 ζ K (R1). By monotonicity for L-if/v-images of L-if/v-subsets, A1 ≤ A2 implies ηA1 ≤ ηA2. On the other hand, by monotonicity for L-if/v-inverse images of L-if/v-subsets, −1 −1 ηA1 ≤ ηA2 implies η ηA1 ≤ η ηA2, but since A1,A2 are constant on Ker(η), −1 −1 we shall get A1 = η ηA1 ≤ η ηA2 = A2.

Hence from the above we shall get that A1 ≤ A2 iff ηA1 ≤ ηA2. ∈ (b) Let A ζ K (R1). By 2.2(1) if A is an L-if/v-ideal of R then η(A) is an L-if/v-ideal of R2 since η is onto homomorphism from R1→R2. Again by 2.2(b), if ηA is an −1 L-if/v-ideal of R2 then η ηA = A is an L-if/v-ideal of R1, since A is constant on Ker(η). Correspondence And Isomorphism Theorems For L-Intuitionistic 1313

Thus A is an L-if/v-ideal of R1 iff ηA is an L-if/v-ideal of R2. 

Corollary 5.2. The complete lattice of all L-if/v-(ideal) sub rings of R1 that are constant on the Ker(η) is isomorphic to the complete lattice of all L-if/v-(ideal) subrings of R2.

Proof. Since the set of all L-if/v-(ideal) subrings of R1 is a complete lattice, the proof follows from the previous theorem. 

6. Second Isomorphism Theorem

Theorem 6.1. For any ring R and for any pair of L-if/v-sub rings A and B of R such that A is an L-if/v-ideal of R and A(0) = B(0), B/(A ∧ B) is L-if/v-weak isomorphic to (A + B)/A, whenever L is a strongly regular complete infinite distributive lattice with CF operator N which is one-one at 0L for L. ∗ ∗ Proof. From 2.1(h), if A is an L-if/v-ideal of R then A is a ideal of R, where A = ∗ ∗ ∗ ∗ ∗ {x ∈ R/µ (x) > 0 and νA(x) < 1}. Hence A ∩ B is an ideal of B , A + B is a A ∗ ∗ ∗ subring of R and A is an ideal of A + B . By the second isomorphism theorem for crisp rings, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ η : B /(A ∩ B )→(A + B )/A , defined by η(b + (A ∩ B )) = b + A is an isomorphism. ∗ ∗ ∗ ∗ ∗ ∗ First we show that (a) (A ∧ B) = (A ∩ B ) (b) (A + B) = A + B . ∈ ∧ ∗ (a) Always x (A B) implies µA∧B(x) > 0 and νA∧B(x) < 1 implying µAx, µ x ≥ µ x ∧µ x = (µ ∧µ )(x) = µ ∧ (x) > 0 and νAx, νBx ≤ νAx ∨νBx = B A B A B ∗A B ∗ ∗ (νA ∨νB)(x) = νA∧B(x) < 1orx ∈ (A ∩B ) which in turn implies that (A∧B) ∗ ∗ ⊆ (A ∩ B ). ∈ ∗ ∩ ∗ On the other hand, since L is strongly regular x A B implies µAx>0 and νAx<1, µ x>0 and νBx<1 which implies µ x ∧µ x>0 and νAx ∨νBx< B A B ∗ ∗ ∗ 1orµ ∧ (x) > 0 and νA∧B(x) < 1 implying x ∈ (A ∧ B) or (A ∩ B ) ⊆ A ∗B ∗ ∗ ∗ (A ∧ B) .Hence (A ∩ B ) = (A ∧ B) . + ∗ { ∈ } (b) Let us recall that (A B) = x R/µA+B(x) > 0 and νA+B(x) < 1 . ∈ + ∗ ∨ ∧ Let x (A B) . Then µA+B(x) = x=y+z(µA(y) µB(z)) > 0 and νA+B(x) = ∧x=y+z(νA(y) ∨ νB(z)) < 1. ∨ ∧ ∈ + Let α = x=y+z(µA(y) µB(z)). There exists y,z R such that x = y z, otherwise α = ∨φ = 0 which is a contradiction. Now if for each y,z ∈ R such that x = y + z, ∧ ∈ µAy µBz = 0 then again α = 0 which is not true. So, there exist y1,z1 R such that + ∧ ≥ ∧ x = y1 z1 and (µA(y1) µB(z1)) > 0. So, α,β α β>0 implies µA(y1)>0, µB(z1)>0, but N is a CF operator with α>0 implies Nα < 1. Therefore µA(y1)>0 implies νA(y1) ≤ Nµ (y1)<1 and µ (z1)>0 implies νB(z1) ≤ NνB(z1)<1 where A B ∗ ∗ ∗ ∗ ∗ x = y1 + z1. Thus x = y1 + z1 with y1 ∈ A , z1 ∈ B or x ∈ A + B . Thus (A + B) ∗ ∗ ⊆ A + B . 1314 G. Vasanti

∈ ∗ + ∗ + ∈ ∗ ∈ ∗ Let x A B . Then x = y z, y A , z B . Hence µA(y) >0, νA(y) < 1, µB(z) > 0 and νB(z) < 1. ∧ ∨ Since L is strongly regular, (µA(y) µB(z)) > 0, (νA(y) νB(z)) < 1 and x = y + z. Hence µ + (x) ≥ µ (y) ∧ µ (z) > 0 and νA+B(x) ≤ νA(y) ∨ νB(z) < 1 A B ∗ A ∗ B ∗ ∗ ∗ ∗ ∗ which implies x ∈ (A + B) . Thus A + B ⊆ (A + B) .Hence (A + B) = A + B . ∗ ∗ ∗ ∗ Consequent of (a) and (b), η : B /(A ∧ B) → (A + B) /A , defined by η(b + (A ∧ ∗ ∗ ∗ B) ) = b + A for each b ∈ B is the isomorphism. Now we show that the above η, in fact, defines an L-if/v-weak isomorphism between the L-if/v-subrings B/(A ∧ B) and (A + B)/A. Observe that, since L is a strongly regular complete infinite-distributive lattice,

∗ ∗ ∗ (1) B/(A ∧ B) : B /(A ∧ B) → L is defined by µ ∧ (b + (A ∧ B) ) ∗ B/(A∗ B) ∗ = ∨µ (b + (A ∧ B) ) and νB/(A∧B)(b + (A ∧ B) ) = ∧νB(b + (A ∧ B) ) B ∗ for each b ∈ B .

∗ ∗ ∗ (2) (A + B)/A : (A + B) /A → L defined by µ + (x + A ) ∗ ∗ (A B)/A ∗ = ∨µ + (x + A ) and ν(A+B)/A(x + A ) = ∧ν(A+B)(x + A ) for each x ∈ (A∗ B) (A + B) .

∗ ∗ ∗ (3) A ∧ B ≤ A implies (A ∧ B) ⊆ A because C ≤ D, x ∈ C implies ≥ ≤ ∈ ∗ µDx µCx>0, νDx νCx<1orx D .

∗ ∗ ∗ ∗ ∗ Now we claim that, η : B /(A ∧ B) → (A + B) /A defined by η(b + (A ∧ B) ) ∗ ∗ = b + A , b ∈ B is the L-if/v-weak isomorphism of B/(A ∧ B) to (A + B)/A. ∗ ∗ ∗ ∗ Let b ∈ B . Let c ∈ (A∧B) be arbitrary. Then z = b+c ∈ b+(A∧B) ⊆ b+B = ∗ ∈ ∗ ∈ + ∧ ∗ + B since b B . Since A(0) = B(0), for each z b (A B) , µ(A+B)z = µ(A+B)(0 z) ≥ µ 0∧µ z = µ 0∧µ z = µ z and ν(A+B)z = ν(A+B)(0+z) ≤ νA0∨νBz = νB0∨νBz A B B B B ∗ ∨ + ∧ ∨ ∗ ≥∨ ∗ = νBz, implying thus µ(A+B)(b (A B) ) = z∈b+(A∧B) µ(A+B)z z∈b+(A∧B) µBz ∨ + ∧ ∗ ∧ + ∧ ∗ ≤∧ + ∧ ∗ = µB(b (A B) ) and ν(A+B)(b (A B) ) νB(b (A B) ). ∗ ∗ ∗ On the other hand since η is defined by η(b + (A ∧ B) ) = b + A for each b ∈ B , is one-one, we get that

+ ∗ ∨ −1 + ∗ + ∧ ∗ (1) µη(B/A∧B)(b A ) = µ(B/A∧B)η (b A ) = µ(B/A∧B)(b (A B) ) ∨ + ∧ ∗ ≤∨ + ∧ ∗ ≤∨ + ∗ = µB(b (A B) ) µ(A+B)(b (A B) ) µ(A+B)(b A ) + ∗ ∧ ∗ ⊆ ∗ = µ((A+B)/A)(b A ), because (A B) A .

∗ −1 ∗ ∗ (2) νη(B/A∧B)(b + A ) = ∧ν(B/A∧B)η (b + A ) = ν(B/A∧B)(b + (A ∧ B) ) ∗ ∗ ∗ = ∧νB(b + (A ∧ B) ) ≥∧ν(A+B)(b + (A ∧ B) ) ≥∧ν(A+B)(b + A ) ∗ ∗ ∗ = ν((A+B)/A)(b+A ), because (A∧B) ⊆ A .Hence B/(A∧B)is L-if/v-weak isomorphic to (A + B)/A.  Correspondence And Isomorphism Theorems For L-Intuitionistic 1315

7. Third Isomorphism Theorem Theorem 7.1. For any L-if/v-sub rings A, B and C of a ring R such that A is an L-if/v- C/A ideal of B and A, B are L-if/v-ideals of C, the L-if/v-sub ring is L-if/v-isomorphic B/A to C/B, whenever L is a strongly regular complete infinite distributive lattice. Proof. (1) Since A is an L-if/v-ideal of B and L is a strongly regular complete infinite ∗ ∗ distributive lattice, by 2.6(2), A is an ideal of B , by 2.8, the L-if/v-quotient B ∗ ∗ ∗ subring exists and by 2.9, (B/A) = B /A . A Since A, B are L-if/v-ideals of C and L is a strongly regular complete infinite ∗ ∗ ∗ distributive lattice, by 2.6(2), A ,B are ideals of C , by 2.8, the L-if/v-quotient C C ∗ ∗ ∗ ∗ ∗ ∗ subrings and exist and by 2.9, (C/A) = C /A and (C/B) = C /B . A B ∗ ∗ C /A ∗ ∗ By Third Isomorphism Theorem for crisp rings, η : → C /B defined by B∗/A∗ ∗ ∗ B ∗ ∗ η(x + A . ) = x + B for each x ∈ C is the ring isomorphism. A∗ (2) B/A is an L-if/v-ideal of C/A as follows: ∗ B B → ∈ ∗ + ∗ Let us recall that : ∗ L is defined by, for each b B , µB/A(b A ) = A A ∗ ∗ ∗ ∗ C C ∨µ (b + A ) and ν (b + A ) = ∧ν (b + A ) and : → L is defined by, B B/A B A A∗ ∈ ∗ + ∗ ∨ + ∗ + ∗ ∧ + ∗ for each c C , µC/A(c A ) = µC(c A ) and νC/A(c A ) = νC(c A ). ∗ ∗ ∗ Now (a) Since B ≤ C on R, for each c ∈ C , µ (c + A ) = ∨µ (c + A ) ≤ ∗ ∗ ∗ B/A ∗ B ∗ ∨µ (c + A ) = µ (c + A ) and νC/A(c + A ) = ∧νC(c + A ) ≥∧νB(c + A ) = C ∗ C/A νB/A(c + A ) or B/A ≤ C/A. (b) Since A is an L-if/v-ideal of both B and C and, L is a strongly regular complete ∗ ∗ infinite distributive lattice, by 2.7, both B/A and C/A are L-if/v-subrings of B /A and ∗ ∗ C /A , respectively. ∗ ∗ ∗ ∗ B C (c) First, since B is a ideal of C , ∗ is a ideal of ∗ . ∗ ∗ A ∗ A ∗ Next, for all a1, a2 ∈ A , b ∈ B and c ∈ C , since B is an L-if/v-ideal of C and A is ∗ ∗ an ideal of B , we get that for each a1, a2 ∈ A , + ∧ + ≤ + + + + ≤ µB(b a1) µC(c a2) µB(c a2 b a1 (c a2)) = µB(bc a1a2) ∨ + ∗ µB(bc A ). Since L is a complete infinite meet distributive lattice, from the above, ∗ ∗ ∗ ∗ µ (b + A ) ∧ µ (c + A ) = ∨µ (b + A ) ∧∨µ (c + A ) B/A ∗ C/A ∗ B ∗ ∗ C ≤∨µ ((b + A ).(c + A )) = µ ((b + A ).(c + A )). B B/A∗ ∗ Similarly, B is an L-if/v-ideal C, A is an ideal of B and L is complete infinite join distributive lattice, imply 1316 G. Vasanti

∗ ∗ ∗ ∗ νB/A(b + A ) ∨ νC/A(c + A ) ≥ νB/A((b + A ) · (c + A )). B C Therefore, is an L-if/v-ideal of . A A C/A C∗/A∗ C∗/A∗ Now by 2.7 and (2) above, : →L is an L-if/v-subring of . B/A B∗/A∗ B∗/A∗ C/A C∗/A∗ (3) By 2.7, 2.8 and (2) above, : →L is defined by: B/A B∗/A∗ ∈ ∗ + ∗ ∗ ∗ ∨ + ∗ ∗ ∗ for each x C , µ C/A ((x A ).(B /A )) = µC/A((x A ).(B /A )) and ∗ B/A∗ ∗ ∗ ∗ ∗ ν C/A ((x + A ) · (B /A )) = ∧νC/A((x + A ) · (B /A )) . B/A C∗/A∗ C∗ On the other hand η : → is defined by: B∗/A∗ B∗  ∗  ∗ ∗ B ∗ for each x ∈ C , η (x + A ) · = x + B . A∗ C/A Now we show that this η defines the necessary isomorphism between and C/B. B/A ∗ ∗ −1 + ∗ + ∗ · B + ∗ · B { + ∗ · + ∗ ∈ Since (a) η is 1-1, η (x B ) = (x A ) ∗ , (b) (x A ) ∗ = (x A ) b A /b ∗ ∗ ∗ A A B } = {xb + A /b ∈ B }, we shall get that  ∗  + ∗ ∨ −1 + ∗ + ∗ · B (i) µη( C/A)(x B ) = µ C/A η (x B ) = µ C/A (x A ) ∗ B/A B/A B/A A  ∗  ∗ B ∗ ∗ = ∨µ (x + A ) · = ∨ ∈ ∗ µ (xb + A ) = ∨ ∈ ∗ (∨µ (xb + A )) C/A A∗ b B C/A b B C ∨ ∗ ∨ ∗ ∨ ∗ = b∈B z∈xb+A µCz = z(∈∪ ∗ xb+A )µCz. b∗∈B ∗ + ∨ + ∨ ∗ On the other hand, µC/B(x B ) = µC(x B ) = z∈x+B µCz.  ∗  + ∗ ∧ −1 + ∗ + ∗ · B (ii) νη( C/A)(x B ) = ν C/A η (x B ) = ν C/A (x A ) ∗ B/A B/A B/A A  ∗  ∗ B ∗ ∗ = ∧ν (x + A ) · = ∧ ∈ ∗ ν (xb + A ) = ∧ ∈ ∗ (∧ν (xb + A )) C/A A∗ b B C/A b B C = ∧b∈B∗ ∧z∈xb+A∗ νCz = ∧z∈(∪ xb+A∗)νCz. b∈B∗

∗ ∗ On the other hand, νC/B(x + B ) = ∧νC(x + B ) = ∧z∈x+B∗ νCz. ∗ ∗ ∗ Now z ∈ x + B implies z ∈ xb + A because 0 ∈ A . Therefore from (i) and (ii) + ∗ ≤ + ∗ + ∗ ≥ + ∗ above, µC/B(x B ) µη( C/A)(x B ) and νC/B(x B ) νη( C/A)(x B ). B/A ∗ ∗ B/A ∗ ∗ On the other hand, z ∈ xb+A for some b ∈ B implies z = xb+a, a ∈ A ⊆ B or ∗ ∗ ∗ z = x+b , b ∈ B , implying z ∈ x+B . Therefore from (i) and (ii) above, µ (x+B ) ∗ ∗ ∗ C/B ≥ µ C/A (x + B ) and νC/B(x + B ) ≤ ν C/A (x + B ). η( B/A) η( B/A) + ∗ + ∗ + ∗ + ∗ Therefore, µη( C/A)(x B ) = µC/B(x B ) and νη( C/A)(x B ) = νC/B(x B )   B/A   B/A C/A C C/A C C/A or η = . Therefore η = . Hence is L-if/v-isomorphic onto B/A B B/A B B/A C/B.  Correspondence And Isomorphism Theorems For L-Intuitionistic 1317

Acknowledgements The author would like to express her gratitude to Professor NistalaV.E.S.Murthy, Andhra university for his continuous help throughout the preparation of this document. She also wants to express her thankfulness to the management of Aditya Institute of Technology and Management, Tekkali.

References [1] Zadeh, L., Fuzzy Sets, Information and Control, Vol. 8, P. 338–353, 1965. [2] Atanassov, K., Intuitionistic Fuzzy Sets, V. Sgurev, Ed., VII ITKR’s Session, June 1983 (Central Sci. and Techn. Library, Bulg. Academy of Sciences, (1984). [3] Gau, W.L. and Buehrer, D.J., Vague Sets, IEEE Transactions on Systems,Man and Cybernetics, Vol. 23, P. 610–614, 1993. [4] Liu, W.J., Fuzzy Invariant Subgroups and Fuzzy Ideals, Fuzzy Sets and Systems, Vol. 8, P. 133–139, 1982. [5] Ren, Y.C., Fuzzy Ideals and Quotient Rings, Fuzzy Math., Vol. 4, P. 19–26, 1985. [6] A. Rosenfeld, Fuzzy Groups, J. Math. Anal. Appl., Vol. 35, P. 512–517, 1971. [7] Banerjee, B. and Basnet, D., Intuitionistic Fuzzy Sub Rings and Ideals, Journal of Fuzzy Mathematics, Vol. 11, No. 1, P. 139–155, 2003. [8] Hur, K., Kang, H.W. and Song, H.K., Intuitionistic Fuzzy Subgroups and Sub Rings, Honam Mathematical Journal, Vol. 25, No. 1, P. 19–41, 2003. [9] Hur, K., Kim, K.J. and Song, H.K., Intuitionistic Fuzzy Ideals and bi-Ideals,Honam Mathematical Journal, Vol. 26, No. 3, P. 309–330, 2004. [10] Jun, Y.B., ztrk, M.A. and Park, C.H., Intuitionistic Nil Radicals of Intuitionistic Fuzzy Ideals and Euclidean Intuitionistic Fuzzy Ideals in Rings, Information Sci- ences, Vol. 177, No. 21, P. 4662–4677, 2007. [11] Wang, J. and Lin, X., Intuitionistic Fuzzy Ideals with Thresholds of Rings, Inter- national Mathematical Forum, Vol. 4, No. 23, P. 1119–1127, 2009. [12] Palaniappan, N., Arjunan, K. and Palanivelrajan, M., A Study on Intuitionistic L-Fuzzy Sub Rings, NIFS 14, Vol 3, P. 5–10, 2008. [13] Meena, K. and Thomas, K.V., Intuitionistic LFuzzy Sub Rings, International Math- ematical Forum, Vol. 6, No. 52, P. 2561–2572, 2011. [14] Murthy V.E.S. Nistala and Vasanti Gopal, Properties of (Inverse) Images of In- tutionistic L-Fuzzy Subsets And Correspondence and IsomorphismTheorems for Intuitionistic L-fuzzy subgroups, Internationa Journal of Computational Cognition (IJCC), Vol. 10, No. 1, P. 18–37, 2011, U.S.A. [15] Vasanti Gopal, More On L-Intuitionistic Zuzzy Or L-Vague Quotient Rings, Ad- vanced Fuzzy Sets And Systems(AFSS), Vol. 16, No. 2, 2013.