J. Comp. & Math. Sci. Vol. 1 (6), 690-695 (2010)
On a certain class of Fuzzy Ideals Leading to Discrete Valuation Rings SOURIAR SEBASTIAN* and GEORGE MATHEW** *Dept of Mathematics, St. Albert’s College Ernakulam **Dept. of Mathematics, BCM College Kottayam ABSTRACT In this paper we characterize valuation rings with the help of fuzzy ideals and discuss the properties of certain class of fuzzy ideals of integral domains which can convert the domains into discrete valuation rings. Key words: Valuations, valuation rings, discrete valuation rings, fuzzy ideals.
1. INTRODUCTION In his classic papers Zadeh13 introduced the notion of a fuzzy subset µ of a given universal set X. Rosenfeld4 used this concept to study the elementary theory of fuzzy groupoids and fuzzy groups. Liu2-3 defined and studied fuzzy ideals in groups and rings. Some works done by the authors in this area are reported in5-12. Integral domains having the property (a) ⊆ (b) or (b) ⊆ (a) for any two elements a and b in the domain are called valuation rings. We here observe that the presence of certain class of fuzzy ideals ensure this total ordering of ideals and thereby the ring becomes a valuation ring. We study the algebra of such fuzzy ideals and prove an equivalent condition for a ring to be a discrete valuation ring.
a function µ : R→[0,1]. The set µ t = {x ∈ R: µ (x) ≥ t} is called a level set of µ for t ∈ [0,1] 2.1 Definition [2] Let µ be a fuzzy subset of R. Then µ is called a fuzzy ideal of R if (i) µ (x + y) ≥ µ(x) ∧ µ(y) (ii) µ (-x) = µ (x) (iii) µ (x y) ≥ µ(x) v µ(y) for all x, y ∈ R 2.2 Definition [3] Let µ and υ be fuzzy subsets of R, then the fuzzy subsets µ∪υ, µ∩υ, µ+υ and µ o υ are defined as follows (µ∪υ)(x)= µ(x) v υ(x) (µ∩υ) (x) = µ(x) ∧ υ(x) (µ+υ) (x) = v {µ(y) ∧ υ(z):x = y+z} (µ o υ) (x) = v {µ(y) ∧ υ(z):x = yz} ∀ x∈ R 2.3 Proposition [3] (i) If µ i, i ∈ I are fuzzy ideals of R then
∩ µ i is a fuzzy ideal of R
i∈I
2.
PRELIMINARIES Algebraic terms and notations used in this paper are as in1.4. Unless otherwise stated all rings are assumed to be commutative with unit element. By a fuzzy subset of R, we mean,
(ii)
(iii)
If µ and υ are fuzzy ideals of R and µ(0) =υ(0), then µ +υ is a fuzzy ideal of R If µ and υ are fuzzy ideals of R then µ o υ is a fuzzy ideal of R
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Souriar Sebastian et al., J. Comp. & Math. Sci. Vol. 1 (6), 690-695 (2010)
691
(ii)
3. LEVEL FUZZY IDEALS
Let x
3.1 Definition. A fuzzy ideal µ of R is called a level fuzzy ideal if every ideal of R is a level set of µ in the sense that given any ideal A of R, we must have A = µ r for some r ∈ [0,1] 3.2 Example. Consider the ring Z9 = {0, 1, 2, 3, 4, 5, 6, 7, 8} under addition and multiplication modulo 9. The ideals of Z9 are {0}, {0, 3, 6} and Z9. The fuzzy ideal {0}→ 1;{ 3,6}→ ½; {1, 2, 4, 5, 7, 8}→ ¼ is a level fuzzy ideal since the above ideals are µ 1, µ 1/2, µ 1/4 respectively 3.3 Example. In a field F, {0} and F are the only ideals. The function µ: F → [0, 1] defined by
1, if x = 0 0, if x ≠ 0
µ (x) =
is a level fuzzy ideal of F 3.4
Example.
Consider the ring
consisting of elements of the form
Z<2>
r where r s
and s are integers having no common factor other than units and s > 0 is odd. We can write r = 2xa with x ≥ 0 and ‘a’ odd. Define x r 2 −1 µ = x . Then µ is a fuzzy ideal on 2 s
r1 2 a r2 2 y b r r = and = . Then 1 + 2 s1 s1 s2 s2 s1 s 2 =
2 x s 2 a + 2 y s1b s1 s 2
If x < y, then
(
r1 r2 2 x s 2 a + 2 y − x s1b + = s1 s 2 s1 s 2
r r 2 x −1 r ∴ µ 1 + 2 = x = µ 1 2 s1 s1 s 2 If x > y, then in a similar way, we get
r1 r2 r + = µ 2 s1 s 2 s2
µ
If
x = y then
r1 r2 u + = 2x+ z s1 s2 v
where z > 0 and u and v are odd
r1 r2 2 x + z − 1 2 x − 1 r ∴µ + = > = µ 1 x+ z x 2 2 s1 s2 s1 r = µ 2 s2
r r r r ∴ µ 1 + 2 = µ 1 ∧ µ 2 s1 s 2 s1 s2 (iii)
r r 2 x+ y − 1 r1 r2 2 x + y ab . = ∴ µ 1 . 2 = s1 s 2 s1 s 2 2 x+ y s1 s 2
Z<2>. The axioms are verified below. Since x + y ≥ x, y; (i)
From the definition of µ it is obvious
−r r = µ s s
that µ
)
r1 r2 . s1 s 2
µ
r r ≥ µ 1 ∨ µ 2 s1 s2
Thus µ is a fuzzy ideal on Z<2>
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Souriar Sebastian et al., J. Comp. & Math. Sci. Vol. 1 (6), 690-695 (2010)
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equivalence relation. The set of all equivalence Consider any non-trivial ideal A of
Z<2> Then A = (q / 1) , q even. Let q = 2xa, x>0 x q 2 −1 and ‘a’ odd. Let t = µ = We claim 2x 1
that A=µ t. Let y∈A, then y =
nq where m is m
odd. Let n = 2hb, h ≥ 0 and ‘b’ is odd. Now y=
2
x+h
x+h
ab 2 −1 2 −1 ∴ µ ( y ) = x +h ≥ x = t m 2 2 ∴ y ∈ µt Hence A⊂µ t. x
On the otherhand, if y∈µ t and
r 2z c y= 1 = , z ≥ 0 and ‘c’ odd s1 s1 Then
2 −1 2 −1 ≥ t = x ∴z ≥ x or z − x ≥ 0 z 2 2 z
µ(y) =
x
Now
2z − x c 2z-x c q 2 z − x 2 x a ∈ Z < 2 > and . = . s1a s1a 1 s1a 1 =
2 z c r1 = s1 s1
∴
r1 ∈ (q / 1) = A, hence µ t ⊂ A. Thus A = µ t s1
Hence µ is a level fuzzy ideal Level fuzzy ideals can exist in rings with zero divisors. The following is an example which supports this observation. 3.5. Example. In the set all rational numbers Q, the relation x ∼ y if x-y ∈Z is an
classes Q/Z is an abelian group with x + y =
x + y . If p is a prime number, the set
m : m , n ∈ z and m , n ≥ 0 is a subgroup n p
R=
of Q/Z. If we define multiplication as x . y =0 for all x , y ∈ R, then R is a ring with zero divisors. Each non zero ideal of R is of the form
1 2 p k −1 Ak= k , k ,...., pk P P
where k is some positive integer and (0)⊂A1⊂A2⊂…⊂R. The fuzzy ideal µ defined by {0}→ 1, A1-{0}→ ½, A2-A1 → →
1 , A3-A2 3
1 …. , R − ∪ Ai → 0 is a level fuzzy ideal i 4
of R The proofs of the following propositions are straight forward and so are omitted. 3.6 Proposition. If µ and υ are level fuzzy ideals of R, then µ∩υ is a level fuzzy ideal. 3.7 Proposition. If µ and υ are level fuzzy ideals of R and µ(0) = υ(0), then µ+υ is a level fuzzy ideal 3.8 Remark. If µ and υ are level fuzzy ideals, µυ need not be a level fuzzy ideal. 3.9 Counter Example. Consider Z9. Define µ and υ by
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Souriar Sebastian et al., J. Comp. & Math. Sci. Vol. 1 (6), 690-695 (2010)
693
µ:{0}→1 {3,6}→1/2 {1,2,4,5,7,8}→1/4 υ:{0}→3/4 {3,6}→1/2 {1,2,4,5,7,8}→1/4 Then µ o υ :{0}→3/4 {1,2,3,4,5,6,7,8}→ ¼ Clearly µ o υ is not a level fuzzy ideal 4. Level Fuzzy ideals and valuation rings Recall that an integral domain R is said to be a valuation ring if for any two ideals A and B, either A ⊆ B or B ⊆ A. In other words, the set of all ideals of R is totally ordered. 4.1 Proposition [1]. For an integral domain V, the following are equivalent (i) V is a valuation ring (ii) If a, b ∈V, then either (a) ⊆ (b) or (b) ⊆ (a) (iii) if K is its quotient field of V, then for every x ∈ K, x ≠ 0, either x ∈ V or x -1 ∈V An abelian group G is said to be ordered if there is defined a total ordering ≤ such that if α, β, γ ∈ G and α ≤ β, then α+ γ ≤ β + γ. Let G be an ordered abelian group and { ∞ } be a set whose single element does not belong to G. Let G = G ∪ { ∞ }. Then the addition operation in G can be extended to G by defining
their sum in G if α , β ∈ G ∞ if α = ∞ or β = ∞
α+ β =
and the ordering in G can be extended to G by defining α ≤ ∞ for all α ∈ G. Then G is a commutative ordered semigroup.
4.2 Definition [1]. Let K be a field. A valuation on K is a map υ from K on to G where G is an ordered abelian group such that (i) υ(a) = ∞ if and only if a = 0 (ii) υ(ab) = υ(a) +υ(b) for all a, b ∈ K (iii) υ(a + b) ≥ min {υ(a), υ(b)} for all a, b, ∈K Further if G is cyclic then υ is called a discrete valuation. A ring with a discrete valuation is called a discrete valuation ring (DVR). 4.3 Theorem . Let R be an integral domain. Then R possesses a level fuzzy ideal µ satisfying the infimum property if and only if R is a discrete valuation ring. Proof. Suppose R possesses a level fuzzy ideal µ satisfying the infimum property. Clearly R is a valuation ring. We prove that R is a PID, and hence is a DVR. Let A be any ideal of R. By hypothesis A = µ t for some t ∈[0, 1]. Since µ satisfies the infimum property, ∃ a ∈µt having the smallest membership value. But then µ t = µ µ(a) ∴ A = µ t = µ µ(a) (1) Again since µ is an level fuzzy ideal, (a) = µ r for some r ∈[0, 1]. Since ‘a’ is having the smallest membership value in (a) it follows that (a) = µ r = µ µ(a) (2) From (1) and (2), we have A = (a) Conversely if R is a DVR, its ideal structure is R ⊃M = (a) ⊃M2= (a2) ⊃………….⊃ (0).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Souriar Sebastian et al., J. Comp. & Math. Sci. Vol. 1 (6), 690-695 (2010)
Choose a1<a2<a3<………….in [0, 1]. Define µ: R→[0,1] by µ(x) = a1 if x ∈ R-(a) = a2 if x ∈(a)-(a2) ----------------------------------------------=Sup an if x ∈(0) Clearly µ is a level fuzzy ideal and every subset of µ (R) = {a1, a2,a3,………} has a smallest element in it. Therefore µ possesses the infimum property. This proves the result 4.4 Remark. If R is a principal valuation domain, we have proved the existence of a level fuzzy ideal having the infimum property. In fact every level fuzzy ideal of R has the infimum property. For, let µ be a level fuzzy ideal of R. Then the ideal structure of R is R ⊃ M = (a) ⊃ (a2) ⊃ ….. ⊃ (0). We claim that µ(x) is a constant, for all x ∈(ai)-(ai+1). Let x ∈(ai)-(ai+1). Then x = r ai, for some r ∈ R. Note that r
∉ (a),
for otherwise x∈(ai+1). It
follows that r is a unit since (a) is a maximal ideal. ∴µ (x) = µ( ai) Also µ is constant in R-M, as all the elements of R-M are units. Therefore the structure of µ is as follows. = t1, x ∈ R-(a) = t2, X ∈ (a)- (a2) --------------------------------------∴µ (R) = { t1< t2<…………}. µ(x)
694
Clearly µ has the infimum property Combining the above theorem with some other equivalent conditions of a discrete valuation ring, we state the following theorem without proof. 4.4 Theorem. Let R be an integral domain having a level fuzzy ideal µ. The following conditions are equivalent. (1) µ possesses the infimum property (2) R is a PID (3) R is Noetherian (4) R is a DVR REFERENCES 1. N S Gopalakrishnan, Commutative Algebra, Vikas Publishing House, New Delhi, (1984). 2. W.J. Liu, Fuzzy Invariant subgroups and Fuzzy Ideals, Fuzzy sets and systems 8, 133-139 (1982). 3. W.J. Liu, Operations on Fuzzy Ideals, Fuzzy Sets and Systems 11, 31-41 (1983). 4. A. Rosenfeld, Fuzzy Groups, J. Math. Anal Appl. 35, 512-517 (1971). 5. Souriar Sebastian and S. Babu sundar, On the chains of level subgroups of homomorphic images and pre-images of fuzzy groups, Banyan Mathematical Journal, 1, 25-34 (1994). 6. Souriar Sebastian and S. Babu sundar, Existence of fuzzy subgroups of level cardinality ℵ0, Fuzzy sets and systems, 67, 365-368 (1994). 7. Souriar Sebastian and S. Babu sundar, Commutative L-fuzzy subgroups, Fuzzy sets and systems, 68, 115-121 (1994). 8. Souriar Sebastian and S. Babu sundar, Generalizations of some results of Das, Fuzzy sets and systems, 71, 251-253 (1995).
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Souriar Sebastian et al., J. Comp. & Math. Sci. Vol. 1 (6), 690-695 (2010)
9. Souriar Sebastian and Thampy Abraham, Fuzzification of Cayleyâ&#x20AC;&#x2122;s and Lagrangeâ&#x20AC;&#x2122;s Theorems, J. Computer and Mathematical Sciences, 1, 41-46 (2009). 10. Souriar Sebastian and Sasi Gopalan, Approximation studies on image enhancement using fuzzy techniques, International Journal of Advanced Science and Technology, 10, 11-26 (2009). 11. Souriar Sebastian and Sasi Gopalan, Modified fuzzy basis function and function
approximation, J. Computer and Mathematical Sciences, 2, 263-273 (2010). 12. Souriar Sebastian and George Mathew, Fuzzy ideals leading to valuation ring, Proceedings of the UGC sponsored national seminar in St. Alberts College, Ernakulam (Aug 2010). 13. L.A. Zadeh, Fuzzy Sets, Inform and Control 8, 338-353 (1965). 14. O. Zariski and P Samuel, Commutative Algebra (Vols 1 and 2) Springer-Verlag, New York (1979).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)