J. Comp. & Math. Sci. Vol.3 (1), 115-120 (2012)
On Fuzzy Ideals and Rings SOURIAR SEBASTIAN’1, MERCY K. JACOB2, V. M. MARY3 and DIVYA MARY DAISE. S4 1
Department of Mathematics, St. Albert’s College, Kochi – 682018, Kerala, India. 2 Department of Mathematics, Toc H Institute of Science and Technology, Aarakunnam - 682313, Kerala, India 3 Department of Mathematics, Maharaja’s College, Ernakulam-682011, Kerala, India 4 Department of Mathematics, Fatima Mata National College, Kollam – 691001, Kerala, India (Received on : 14th February, 2012) ABSTRACT In this expository article we discuss fuzzy rings, fuzzy [left, right] ideals and fuzzy maximal [prime] ideals of a general ring R and present some results regarding these concepts. Keywords: Fuzzy ring, fuzzy ideal, fuzzy maximal ideal, fuzzy prime ideal.
1. INTRODUCTION
2. BASIC CONCEPTS
The innovative works of Zadeh10 and Rosenfeld7 led to the fuzzification of algebraic structures. Several mathematicians like Kuroki2, Wan-Jin-Liu9, Malik and Mordesan3 and Mukherjee and Sen5 have worked in this area and obtained significant results. In this expository article, we discuss fuzzy rings, fuzzy [left, right] ideals and fuzzy maximal [prime] ideals of a general ring and present some results regarding these concepts. Terms and results from ring theory used in this work are as in J. B. Fraleigh1.
Throughout this work R denotes a ring (R, +, °) unless otherwise stated and ℤ, ℚ, ℝ and ℂ denote the ring of integers, rational numbers, real numbers and complex numbers respectively. By a subring S of R we mean a non-empty subset of R which itself is a ring with respect to the same operations as in R. For example ℤ is a subring of ℚ; which is a subring of ℝ; which in turn is a subring of ℂ. For any positive integer n, nℤ denotes the set { 0, ±n, ±2n, ±3n,…} obtained by multiplying the elements of ℤ by n.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
116
Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 115-120 (2012)
A non-empty subset I of a ring R is said to be a right ideal of R if (i) a – b ∈ I, ∀ a, b ∈ I and (ii) ar ∈ I, ∀ a ∈ I and r ∈ R. In other words, a right ideal of R is a subgroup of R under addition which absorbs right multiplication by elements of R. Similarly, by a left ideal of R we mean an additive subgroup of R which absorbs left multiplication by elements of R. We say that I is an ideal of R if it is both a right ideal and a left ideal. That is, if (i) a – b ∈ I, ∀ a, b ∈ I and (ii) ar, ra ∈ I, ∀ a ∈ I and r ∈ R. It is well known that every ideal is a subring; but the converse is not true. In any ring R, we have two trivial ideals, viz; {0} and R itself. Any ideal different from these two is called a proper ideal (or a non – trivial ideal). For example, the set 2ℤ = { 0, ±2, ±4, ±6,...} is a proper ideal of ℤ. If R is a field, then it has no proper ideals. An ideal M(≠R) of a ring R is said to be a maximal ideal if we cannot squeeze in an ideal between them. That is, if there does not exist an ideal I of R such that M ⊂ I ⊂ R. Also, an ideal P(≠R) is called a prime ideal if ab ∈ P ⇒ either a ∈ P or b ∈ P. If R is a commutative ring with unity, then every maximal ideal of R is a prime ideal also. For example, 2ℤ, 3ℤ, 5ℤ, 7ℤ, …………… are all maximal ideals as well as prime ideals of ℤ. But 4ℤ is a maximal ideal which is not a prime ideal in the ring 2ℤ, which is a commutative ring without unity.
3. FUZZY RINGS AND FUZZY IDEALS A fuzzy set on a ring R (or a fuzzy subset of R ) is a function A : R [0, 1]. Ordinary subsets of R will be called crisp subsets. For any α ∈ [0, 1] and fuzzy set A on R, the set αA = {x ∈ R | A(x) ≥ α } is called the α - level set of A. Allowing α to vary over [0, 1] we get all α - level sets of A, which are all crisp subsets of R. If A, B are fuzzy sets on R, then A ⊆ B if A(x) ≤ B(x), ∀ x ∈ R and A = B if A(x) = B(x), ∀ x ∈ R. Also, their standard union A ∪ B, standard intersection A ∩ B and standard are the fuzzy subsets of R complement A defined by (A∪B)(x) = max {A(x), B(x)}, = 1 – A(x), (A∩B(x) = min { A(x), B(x) }, A ∀ x ∈ R. If {Ai : i ∈ I } is an arbitrary collection of fuzzy sets on R, where I is any index set, then ∪i∈I Ai and ∩i∈I Ai are defined by (∪i∈I Ai)(x) = supi∈I Ai(x) and (∩i∈I Ai)(x) = infi∈I Ai(x), ∀ x ∈ R. 3.1. Definition5. A fuzzy set A on R is said to be a fuzzy ring on R if for every x, y ∈ R: (i) A(x-y) min {A(x), A(y)}; and (ii) A(x.y) min { A(x), A(y) }. Sometimes, when we want to emphasize the role of the ring, we shall say that A is a fuzzy subring of R. 3. 2. Example. Define A : ℤ [0, 1] by A (x) = 0.8, if x = 0 and = 0.1, if x ≠ 0. It can be easily verified that A is a fuzzy ring on ℤ. 3. 3. Example. Let F be any field and R = (K[x, y, z], +, °) be the ring of polynomials
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 115-120 (2012)
117
in three indeterminates x, y and z over F. Define A : R [0, 1] by µ(x) = 1, if P ∈ K[x]; = 0.45, if P ∈ K[x, y] – K[x]; = 0.25, if P ∈ K[x, y, z] – K[x, y]. Then A is a fuzzy ring on R.
3.7. Proposition4. Let R1, R2 be rings, f : R1 R2 be a ring homomorphism, A be a fuzzy ring on R1 and B be a fuzzy ring on R2. Then f(A) is a fuzzy ring on R2 and f-1(B) is a fuzzy ring on R1 █
3.4. Proposition. The characteristic function χR of a ring R is a fuzzy ring on R. Proof: Straight forward █
3. 8. Definition4. A fuzzy ring A on a ring R is said to be a fuzzy left ideal if A(x°y) ≥ A(y), ∀ x, y ∈ R and a fuzzy right ideal if A(x°y) ≥ A(x), ∀ x, y ∈ R A is called a fuzzy ideal if it is both a fuzzy left ideal and a fuzzy right ideal. In other words, a fuzzy set A on R is a fuzzy ideal if, ∀ x, y ∈ R (i) A(x-y) ≥ min { A(x), A(y) }, and (ii) A(x°y) ≥ max { A(x), A(y) }.
3.5. Proposition. Let S be a non – empty subset of R. Then S is a subring of R if, and only if, its characteristic function χS is a fuzzy ring on R. Proof: Straight forward █ For a fixed element k ∈ [0,1], the fuzzy set k on R defined by k (x) = k, ∀ x ∈ R, is called a constant fuzzy set on R. Obviously all constant fuzzy sets on R satisfy both the axioms of a fuzzy ring. Hence all constant fuzzy sets on R are fuzzy rings on R. Now, turning to non-constant fuzzy sets on R, we get the following proposition. 3. 6. Proposition7. A non-constant fuzzy set A on R is a fuzzy ring on R if, and only if, αA is a subring of R, ∀ α∈ Im(A) █ If R1, R2 are rings, f : R1 R2 is a function, A is a fuzzy set on R1 and B is a fuzzy set onR2, then the image of A under f is the fuzzy set f(A) on R2 defined by f(A)(y) = sup { A(x) | x ∈ f-1(y) } =0, if f-1(y) = ϕ,∀ y ∈ R2. The pre-image of B under f is the fuzzy set f-1(B) on R1 defined by f-1(B)(x) = B[f(x)], ∀ x ∈R1.
3.9. Example. Consider the ring ℤ9 = { 0, 1, 2, …, ,8 } with respect to the operations +9 and ×9. Define A : ℤ9 [0,1] by A(x) = 0.9, if x = 0 = 0.4, if x = 3 or 6 = 0.1, otherwise. Then A is a fuzzy ideal on ℤ9. 3.10. Example. Let R denote the polynomial ring F[x, y] over a field F. Define the fuzzy set A on R by A(P) = 1, if P ∈<x2, xy> = 0.4, if P ∈ < x2, xy, y2> - <x2, xy> = 0.3, otherwise. It can be easily verified that A is a fuzzy ideal on R. 3.11. Proposition4. A fuzzy ring on R is a fuzzy [left, right] ideal iff Aα is a [left, right] ideal of R, ∀ α ∈ [0,1] █
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
118
Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 115-120 (2012)
We give below some properties of fuzzy [left, right] ideals. 3.12. Proposition5. If Ai; i ∈I, are fuzzy [left, right] ideals of R, then ∩i∈I Ai is a [left, right] ideal of R █ If A is a fuzzy ring on R, we shall use the notation A0 for { x ∈ R | A(x) = A(0) }. 3.13. Proposition4. If A, B are fuzzy ideals of R, then (i) A(0) ≥ A(x) and A(-x) = A(x), ∀ x ∈ R (ii) If R has multiplicative identity 1, then A(1) ≤ A(x), ∀ x ∈R (iii) For x, y ∈ R A(x-y) = A(0) ⇒ A(x) = A(y) (iv) A0 is an ideal of R (v) A0 ∩ B0 ⊆ (A ∩ B)0 █ 4. FUZZY MAXIMAL IDEALS In the crisp case, the concept of maximal ideal is central to the applications of commutative ring theory to algebraic geometry. Therefore they form an important class of ideals in commutative ring theory. In this section, we discuss its fuzzy counterpart, viz; fuzzy maximal [left, right] ideals. Fuzzy maximal ideals are defined and studied by Malik and Mordeson3. They proved that fuzzy maximal ideal A of R cannot be defined as a fuzzy ideal different from the characteristic function of R such that B is a fuzzy ideal of R and A ⊂ B ⊆ χR ⇒ B = χR Instead they approached the notion of fuzzy maximal ideals through fuzzy maximal left [and right] ideals.
4.1. Definition3. Let A be a non-constant fuzzy left [right] ideal of R. Then A is called a fuzzy maximal left [right] ideal of R if for any fuzzy left [right] ideal B of R A ⊂ B ⇒ A0 = B0 or B = χR. In the following propositions, we give some properties of fuzzy maximal left ideals. Similar results hold for fuzzy right ideals also. 4.2. Proposition3.If A is a fuzzy maximal left ideal of R then (i) A(0) = 1 (ii) |Im(A)| = 2, and (iii) A0 is a maximal ideal of R █ 4.3. Proposition3. If A is a fuzzy left ideal of R such that A0 is a maximal left ideal of R and A(0) = 1, then A is a fuzzy maximal left ideal of R █ 4.4. Proposition3. An ideal I (≠R) is a maximal left ideal of R if, and only if, χI is a maximal left ideal of R █ 4.5. Proposition3. Let R be a ring with identity and A be a non-constant fuzzy left ideal of R. Then there exist a fuzzy maximal left ideal B of R such that A ⊆ B █ 4.6. Remark. It may be recalled that an element α in a complete lattice L is said to be a dual atom if α ≠ 1 and there does not exist an element β ∈ L such that α < β < 1. In other words, a dual atom α ∈ L is a maximal element in L – {1}. Swamy and Swamy8 gave the following characterization of maximal fuzzy ideals in L-fuzzy case, where the membership set [0, 1] is replaced by a non-trivial complete lattice L in which the infinite meet distributive law
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 115-120 (2012)
a ∧ (∨s∈S s) = ∨s∈S(a ∧ s) holds for any a∈ L and S ⊆ L. 4.7. Theorem8. Let A be a fuzzy set on R. Then A is a fuzzy maximal ideal of R if, and only if, there exist a maximal ideal M of R and a dual atom α ∈ L such that A(x) = 1, if x ∈ M, and = α, otherwise∎
119
A . B ⊆ F ⇒ either A ⊆ F or B ⊆ F. Recall that the notation a|b means that a divides b or equivalently, b is a multiple of a. 5.3. Example. Consider the ring ℤ of all integers. Define F : ℤ [0, 1]by F(x) = 1, if 5|x = 0.2, otherwise. Then F is a fuzzy prime ideal of ℤ.
4.8. Remark. The above theorem says that there is a 1-1 correspondence between fuzzy maximal ideals of R and pairs (M, α) where M is a maximal ideal of R and α is a dual atom in L. if L = [0,1], then there is no dual atom in L. Hence, in this case, R has no fuzzy maximal ideals.
5.4. Proposition5. Let F be a fuzzy prime ideal of a ring R. Then F0 = { x ∈ R | F(x) = F(0) } is a prime ideal of R █ For any fuzzy ideal F of ℤ, F0 = { x ∈ ℤ | F(x) = F(0) } is an ideal of ℤ; and hence we can find a positive integer n such that P0 = n ℤ.
5. FUZZY PRIME IDEALS
5.5. Proposition5. Let F be a fuzzy ideal of ℤ with F0 = n ℤ, where n is a positive integer. Then F can take atmost r values, where r is the number of distinct positive divisors of n █ Now we proceed to give the characterization of all fuzzy prime ideals of ℤ given by Mukherjee and Sen5.
Due to the importance of prime ideals in classical ring theory, fuzzy primeness has been given much attention. Fuzzy prime ideals were defined and studied by Mukherjee and Sen5. In this section, we discuss the notion of fuzzy prime ideals as defined by Mukherjee and Sen and present some of their important properties. We also present a characterization of all fuzzy prime ideals of ℤ obtained by Mukherjee and Sen; and its corrected version given by Swamy and Swamy8. 5.1. Definition4. For fuzzy sets A and B on a ring R we define the product A . B as the fuzzy set on R given by (A . B)(x) = supx=yz{min(A(y), B(z))} = 0, if x is not expressible as x = yz. 5.2. Definition4. A non-constant fuzzy ideal F of a ring R is said to be a fuzzy prime ideal if for any fuzzy ideals A and B of R,
5.6. Theorem. Let P be a non-null (i.e. P0 ≠ {0}) fuzzy prime ideal on ℤ. Then P has two distinct values. Conversely, if P is a fuzzy subset of ℤ such that P(n) = α1, when p|n and P(n) = α2, otherwise, where p is a fixed prime number and α1 > α2, then P is a nonnull fuzzy prime ideal on ℤ █ Later Swamy and Swamy8 gave a characterization of fuzzy prime ideals of a general ring R, in the L-fuzzy case, as two valued, the top value being 1. Using that characterization, they corrected the above theorem as follows.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
120
Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.3 (1), 115-120 (2012)
5.6. Theorem. A fuzzy set P on ℤ is a fuzzy prime ideal if, and only if, it is given by P(n) = 1, if p|n = α, otherwise where p is a prime integer or zero and α < 1 █ 5.7. Remark. For a fixed prime number p, we get the prime ideal p ℤ of ℤ. Fixing α in [0, 1) we get a unique fuzzy prime ideal of ℤ as in the above theorem. We may allow p to vary over all positive prime numbers and α to take various values in [0, 1). Also, the fuzzy subset P of ℤ given by P(n) = 1, if n = 0 = α, otherwise where α <1, is a fuzzy prime ideal of ℤ. This provides us with uncountably many fuzzy prime ideals of ℤ. But each of them is twovalued. Further, the theorem asserts that these are the only fuzzy prime ideals of ℤ. 6. REFERENCES. 1. Fraleigh J. B, A First Course In Abstract Algebra, Narosa Publishing House, New Delhi (1998).
2. Kuroki. N, On Fuzzy Ideals And Fuzzy Bi-ideals In Semigroups, Fuzzy Sets And Systems, 5, 203-215 (1981). 3. Malik D. S. & Mordes on J. N, Fuzzy Maximal, Radical and Primary Ideals of A Ring, Inform. Sci., 53, 237-250 (1991). 4. Mordeson J. N & D. S. Malik, Fuzzy Commutative Algebra, World Scientific, Singapore (1998). 5. Mukherjee T. K & Sen M. K, On Fuzzy Ideals of A Ring I, Fuzzy Sets and Systems, 21, 99-105 (1987). 6. Naseem Ajmal & K. V. Thomas, The Lattices Of Fuzzy Ideals of A Ring, Fuzzy Sets and Systems, 74, 371-379 (1995). 7. Rosenfeld. A, Fuzzy Groups, J. Math. Anal. Appl., 35, 512-517 (1971). 8. Swamy U. M & Swamy K. L. N, Fuzzy Prime Ideals of Rings, J. Math. Anal. Appl., 134, 94-103 (1988). 9. Wang-Jin-Liu, Operations On Fuzzy Ideals, Fuzzy Sets and Systems, 11, 3141 (1983). 10. Zadeh L. A, Fuzzy Sets, Inform Control, 8, 338 – 353 (1965).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)