Cmjv02i01p0051 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

Equivalence of Certain Class of Fuzzy Ideals and Real Valuation Rings SOURIAR SEBASTIAN1 and GEORGE MATHEW2 Dept of Mathematics, St. Albert’s College, Ernakulam, Kochi – 682018, Kerala, India e- mail : souriarsebastian@gmail.com Dept of Mathematics, B.C.M College, Kottayam-686001, Kerala, India e- mail : gmathew5616x@gmail.com ABSTRACT We have introduced the notion of certain class of fuzzy ideals called valuation fuzzy ideals in16 and derived the conditions required for the existence of such ideals. We also established an order preserving correspondence between such ideals and valuations of a valuation ring. In this paper we continue this investigation. Even though the existence of a valuation fuzzy ideal guarantees that the ring is a valuation ring, it may either be a real valuation ring or a non-real one. We therefore introduce another class of fuzzy ideals called ℝ-valuation fuzzy ideals and prove that an integral domain is a real valuation ring if and only if it possesses a ℝ-valuation fuzzy ideal. We also represent the value group of real valuation rings in terms of ℝ-valuation fuzzy ideals and discuss a method of constructing such ideals in a DVR. Key words: Valuations, value group, valuation rings, discrete valuation rings(DVR), prime ideals, fuzzy ideals, valuation fuzzy ideals.

1. INTRODUCTION The notion of ‘fuzzy set’ was introduced by Zadeh17. Given a universal set X, a function µ: X ⟶[0,1] is called a fuzzy subset of X . Rosenfeld6 used this idea to define fuzzy groupoids and fuzzy groups. Liu [3-4] defined and studied fuzzy ideals in groups and rings. Some works done in this area by the authors are reported in7-16. In16 we have introduced the notion of certain

class of fuzzy ideals called valuation fuzzy ideals and derived the conditions required for the existence of such ideals. We also established an order preserving correspondence between such ideals and valuations of a valuation ring. In this paper we continue this investigation. Even though the existence of a valuation fuzzy ideal guarantees that the ring is a valuation ring, it may either be a real valuation ring or a non-real one. We

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

therefore introduce another class of fuzzy ideals called ℝ-valuation fuzzy ideals and prove that an integral domain is a real valuation ring if and only if it possesses an ℝ-valuation fuzzy ideal. We also represent the value group of real valuation rings in terms of ℝ-valuation fuzzy ideals and discuss a method of constructing such ideals in a DVR.

An abelian group (G,+) is called an ordered abelian group if there is defined a total ordering ≤ on G such that હ ≤ ઺ ⟹ હ + ઻ ≤ ઺ + ઻ ∀ હ , ઺, ઻ ∈ G. Let G be an ordered abelian group and ‘∞’ be an element which does not belong to G. Let ࣡ = G∪{∞}. Then the addition operation on G can be extended to ࣡ by defining

2. BASIC CONCEPTS

α+ β = 

Throughout this paper, R denotes a commutative ring with unit element, unless otherwise stated. Algebraic terms and notations used here are either standard or as in2 and 18. For a fuzzy subset µ of R and t ∈ [0,1] the set µt = {x∈R : µ(x) ≥ t} is called the level set of µ corresponding to t. It may be recalled that a fuzzy subset µ of R is called a fuzzy ideal[3] of R if (i) µ (x + y) ≥ µ(x) ∧ (y), (ii) µ(-x) = µ(x) and (iii) µ (xy) ≥ µ(x) ∨ µ(y) for all x, y ∈ R. Here ∧ and ∨ denote the max (or sup) and min (or inf) operators respectively. We denote the set { x ∈ R : µ(x) = µ(0) } by µ∗ . An integral domain R is said to be a valuation ring if for any two ideals A and B of R, we have either A ⊆ B or B ⊆ A. That is, if ‘⊆’ is a total order on the set of all ideals of R. If a, b ∈ R, by saying that a/b ∈ R, we mean a = rb for some r ∈ R. Therefore (a) ⊆ (b) ⇔ a/b ∈ R. 2.1. Proposition [18] For an integral domain R the following are equivalent (i) R is a valuation ring (ii) If a, b ∈ R, then either (a) ⊆ (b) or (b) ⊆ (a) (iii) If K is the quotient field of R, then for every x (≠0) ∈ K, either x ∈ R or x -1 ∈R ∎

 their sum in G, if α , β ∈ G ∞, if α = ∞ or β = ∞

and the ordering in G can be extended to ࣡ by defining α ≤ ∞ for all α∈ G. Then ࣡ becomes a commutative ordered semi-group. A natural example for this is obtained by taking G = Z, the group of integers under addition. In this case ࣡ = Z ∪ {∞}. 2.2. Definition[18]. Let K be a field. A valuation on K is a map υ from K on to ࣡, where ࣡ is obtained from an ordered abelian group G as above, satisfying the following conditions (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 In this case we say that G is the value group of υ. The following proposition gives the relationship between the notions of “valuation” and “valuation ring”. 2.3. Proposition[18]. If υ is a valuation on a field K, then V = {a ∈ K: υ(a) ≥ 0} is a valuation ring in K. Conversely, if V is a valuation ring with quotient field K, then there is a valuation υ on K with value group G = ॶ/U where ॶ is the multiplicative group of all non zero elements of K and U is the

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

subgroup consisting of all unit elements of V and that V = {a ∈ K : υ(a) ≥ 0} ∎ Let υ be a valuation on a field K with value group G and valuation ring V. If ℝ represents the additive group of all real numbers and G ⊆ ℝ , then V is called a real valuation ring and the corresponding valuation υ is called a real valuation. If G = Z, the group of integers, then R is called a discrete valuation ring (DVR). If V is a DVR, then its ideal structure is V ⊃ (a) ⊃ (a2) ⊃ ………..⊃ (0) where ‘a’ is the single irreducible element of V. 3. FUZZY IDEALS AND REAL VALUATION RINGS 3.1. Theorem . Let V be an integral domain. Then V is a real valuation ring if and only if ∃ a fuzzy ideal µ satisfying the following conditions. (i) µ * = {0} (ii) µ(x) + µ(y) = µ(xy) + µ(x) µ(y) ∀ x, y ∈ V and (iii) µ (y) ≤ µ(x) ⇒ x/y ∈ V, ∀ x, y ∈ V Proof. Let V be a real valuation ring. Then ∃ a valuation υ on its quotient field K with value group G ⊆ R. Define µ: V → [0, 1] by µ(x) = 1-2- υ (x) . Then from the properties of valuation, it can easily be verified that µ(x + y) ≥ µ(x) ∧ µ(y), µ(xy) ≥ µ(x) ∨ µ(y) and µ(-x) = µ(x).

= 1 – 2-υ(x) 2-υ(y) + 1 – 2-υ(x) – 2-υ(y) + 2-υ(x) 2-υ(y) = (1 – 2-υ(x)) + (1 – 2-υ(y)) ∴ µ(x) + µ(y) = µ(xy) + µ(x) µ(y). This proves (ii) Again, µ(0) = 1 – 2-υ(0) = 1 – 2-∞ = 1. Also, x ∈µ 1 ⇒ µ(x) = 1 ⇒ 1 – 2-υ(x) = 1 ⇒ υ(x) = ∞ ⇒ x = 0. ∴ µ * = {0}. This proves (i). Finally, for x, y∈V, µ(y) ≤ µ(x) ⇒ υ(y) ≤ υ(x) ⇒ υ(x/y) = υ(x) –υ(y) ≥ 0. ∴ x/y ∈ V. This proves (iii). Conversely suppose ∃ a fuzzy ideal µ on V satisfying condition (i) and (ii) and (iii). Define υ(x) = - log2 (1-µ(x)). We claim that υ(x) = ∞ ⟺ x = 0. We have from the definition of υ, υ(0) = - log2 (1-µ(0)). But, by condition (ii), µ(0) + µ(0) = µ(0) + µ(0). µ(0). Therefore µ(0) (µ(0)-1) = 0. Since µ(0) ≠ 0, we have µ(0) = 1. Hence υ(0) = ∞. Also, υ(x) = ∞ ⇒ - log2 (1 – µ(x)) = ∞ ⇒ 1- µ(x) = 0 ⇒ µ(x) = 1. By condition (i), x = 0. ∴ υ(x) = ∞ ⟺ x = 0. Again, υ(xy) = -log2 (1 – µ(xy)) = -log2 (1 – µ(x) + µ(y) – µ(x) µ(y)) ( by condition (ii) ) = -log2 (1 – µ(x) (1 – µ(y)) = υ(x) + υ(y) Also υ(x + y) ≥ υ(x) ∧ υ(y). Therefore υ is a real valuation on the quotient field K and V is a subring of its valuation ring. In fact, the valuation ring of υ is

{x/y ∈ K : µ(y) ≤ µ(x)} .

But

Therefore µ is a fuzzy ideal. In order to prove condition (ii), we have

by condition (iii), this is equal to V. ∎

µ(x) + µ(y) = (1 – 2-υ(x)) + (1 – 2-υ(y)) and µ(xy) + µ(x) µ(y) = 1 - 2-υ(xy) + (1 - 2-υ(x))(1 - 2-υ(y))

3.2. Definition [16]. A fuzzy ideal µ of a ring R is said to be a valuation fuzzy ideal if

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

µ(x) < µ(y) ⇒ (y) ⊂ (x) and µ(x) = µ(y) ⇒ (y) = (x) 3.3. Definition. A fuzzy ideal µ of a ring R is said to be an ℝ-valuation fuzzy ideal if (i) µ * = {0}, (ii) µ(x) + µ(y) = µ(xy) + µ(x) µ(y) and (iii) µ(y) ≤ µ(x) ⇒ x/y ∈ V In terms of the ℝ-valuation fuzzy ideal, we can restate theorem 3.1 as follows. An integral domain V is a real valuation ring if and only if it possesses an ℝ-valuation fuzzy ideal. 3.4. Proposition. Every ℝ-valuation fuzzy ideal is a valuation fuzzy ideal. Proof. Suppose µ is an ℝ-valuation fuzzy ideal of an integral domain V. Let x, y ∈ V and µ(x) < µ(y). Then by condition (iii), y/x ∈ V. ∴y = r x, r ∈V. We prove that r is a non-unit. We have µ(r x) = µ(r) + µ(x) – µ(r). µ(x) by condition (ii). ∴µ(y) = µ(r) + µ(x) – µ(r) . µ(x), hence µ(r) – µ(r) µ(x) = µ(y) – µ(x) >0. ie, µ(r) (1 – µ(x)) > 0. Since both the factors on the L.H.S are non-ve, we have µ(r) > 0. We claim that x ∈ V is a unit ⇔ µ(x) = 0 Note that by condition (ii), we have .

µ(1) + µ(1) = µ(1) + µ(1). µ(1). ∴ µ( 1) (1 – µ(1)) = 0. But by condition (i), µ(1) ≠ 1. ∴µ(1) = 0. If x is a unit, then xy = 1 for some y∈V. But then µ(1) = µ(xy) ≥ µ(x) or µ(x) ≤ µ(1). It follows that µ(x) = µ(1) = 0. On the other hand, if µ(x) = 0 for x ∈ V, then µ(x) = µ(1). By condition (iii), 1/x ∈ V. ∴x is a unit. Since µ(r) > 0, it follows that r is a non-unit. Therefore (y) ⊂ (x). Again, µ(x) = µ(y) ⇒ x/y ∈ V and y/x ∈ V by condition

(iii). ∴(y) = (x). Thus µ(x)<µ(y) ⇒ (y) ⊂ (x) and µ(x) = µ(y) ⇒ (y) = (x). ∴µ is a valuation ideal. ∎ It is not true that every valuation fuzzy ideal is an ℝ-valuation fuzzy ideal. The following example substantiates this remark. 3.5. Example. Let Z(2) represents the ring consisting of all rational numbers of the form r/s where r, s ∈ Z and s is odd. Then Z(2) is called the localization of the ring of integers Z. Its ideal structure is Z(2) ⊃ (2/1) ⊃ (22/1) ⊃ (23/1) ⊃…………⊃ (0), where S is the set of all odd integers. Define µ: Z(2) ⟶ [0,1] by µ(x) = 0, if x ∈ Z(2) – (2/1) = n/(n+1), if x ∈ (2n/1) – (2n+1/1), n = 1, 2, …. = 1, if x = 0 Then µ is a valuation ideal. Let x = 4/3 and y = 12/5. Then x, y ∈ (22/1) – (23/1) ∴ µ(x) = µ(y) = 2/3. xy = 48/15 = 16/5 ∈ (24/1) – (25/1). Now µ(x) + µ(y) = 4/3 = 60/45 and µ (xy) + µ(x) µ(y) = 4/5 + 4/9 = 56/45. ∴µ(x) + µ(y) ≠ µ (xy) + µ(x) µ(y). Hence µ is not an ℝ-valuation fuzzy ideal. ∎ We have seen from theorem 3.1 that ℝ-valuation fuzzy ideals and real valuations are equivalent for an integral domain. We therefore denote the valuation equivalent to an ℝ-valuation fuzzy ideal µ by υµ and the ℝ-valuation fuzzy ideal equivalent to a valuation υ by µυ. We shall extend this idea to the context of quotient rings. For this, we need the following results. 3.6. Proposition[18]. If υ is a valuation on an integral domain V and P is a prime ideal of V then the quotient ring V/P is an integral domain and the map υ/P on V/P defined by

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

υ/P (x+P) = ∨{υ(x+z) : z ∈ P} is a valuation on V/P. ∎ 3.7. Proposition[5]. If µ is a fuzzy ideal on R and P is a prime ideal of R then the map µ/P defined on the quotient ring R/P by (µ/P) (x+P) = ∨{µ(x+z) : z ∈ P} is a fuzzy ideal on R/P. ∎ Let µ be an ℝ-valuation fuzzy ideal on an integral domain V and υµ be the equivalent valuation. If P is a prime ideal of V, then there are two quotient fuzzy ideals on V/P (i) µ/P, the fuzzy ideal of µ relative to P and (ii) µ (υµ / P) , the ℝ-valuation fuzzy ideal equivalent to the valuation υµ/P. In the following proposition, we prove that either of them are equal. 3.8. Proposition. Let µ be an ℝ-valuation fuzzy ideal on an integral domain V and υµ be the equivalent valuation. Let P be a prime ideal of V. Then µ (υµ / P) = µ/P. Proof. We have υµ(x) = - log2(1-µ(x)). Also by definition (υµ/P) (x+P) = ∨{υµ(x+z) : z ∈ P}

µ (υµ / P) (x + P) = 1 − 2

− (υµ / P) (x +Ρ)

=1− 2

−∨{υµ (x + z ): z∈P}

=1− 2

−∨{ − log 2 (1− µ(x + z )): z∈P}

= 1 − 2∨{log2 (1−µ(x + z )):z∈P} Since the function f(x) = log2 x is strictly increasing,

∨{log 2 (1 − µ(x + z)) : z ∈ P} = log 2 (∨{1 − µ(x + z) : z ∈ Ρ})

It follows that

µ ( υµ / P) (x + P) = 1 − ∨{1 − µ(x + z) : z ∈ P} = ∨{µ(x + z) : z ∈ Ρ}

∴

µ (υµ / Р)

=(µ/P)(x+P) = µ/P. ∎

4. VALUE GROUP FROM ℝ-VALUATION FUZZY IDEALS 4.1. Proposition. Let V be an integral domain having an ℝ-valuation fuzzy ideal µ. Then V is a real valuation ring with value group   1 − µ(y)   P= log 2  : x,y ∈ V; x ≠ 0, y ≠ 0  .     1-µ(x)    Proof. In theorem 3.1 we have proved that an integral domain V having an ℝ-valuation fuzzy ideal is a real valuation ring and we know that for a valuation ring V, the value group is G = ॶ/U where ॶ is the multiplicative group of all non-zero elements of its quotient field K and U is the subgroup consisting of all non units of V. Therefore we need only to prove that P is a sub group of ℝ and that P is isomorphic to G. Since µ is an ℝ-valuation fuzzy ideal, we have µ * = {0}, µ(x) + µ(y) = µ(xy) + µ(x) µ(y) and µ(y) ≤ µ(x) ⇒ x/y ∈ V for all x,y ∈ V. In order to prove that P is a subgroup of ℝ, let

1 − µ(y)    1-µ(x) 

X, Y ∈ P, X = log2 

1 − µ(y')  and Y = log 2  . 1 − µ(x')  Then X + Y = log2 1 − µ(y)   1-µ(x)  + log 2  

log 2

 1 − µ(y')  1 − µ(x')  =   (1 − µ(y) )(1 − µ(y') )

(1 − µ(x) )(1 − µ(x') )

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

 1 − ( µ ( y ) + µ ( y ') − µ ( y ) µ ( y ') )     1 − ( µ ( x ) + µ ( x ') − µ ( x ) µ ( x ') ) 

= log2

= lo g

 1 − µ ( y y ')   1 − µ ( x x ')   

= log2

∴X + Y∈ P . Again, - X = -log2 P. ∴P is a subgroup of ℝ. In order to prove that G is isomorphic to P, note that any element of G is of the form (x/y)U . Consider the map from G into P defined by

1 − µ (y)  (x/y)U → log 2   1 − µ (x)  The

map

well defined , for, if (x/y)U = (x'/y')U , then xy'/yx' ∈ U. ∴ xy'/yx' = u where u is a unit in V. ∴xy' = uyx', hence µ(xy') = µ(yx'). ∴µ(x) + µ(y') – µ(x) µ(y') = µ(y) + µ(x') – µ(y) µ(x'). Hence 1 – µ(x) – µ(y') + µ(x) µ(y') = 1µ(y) - µ(x') + µ(y) µ(x'). ∴(1 – µ(x)) (1 – µ(y')) = (1 – µ(y))(1-µ(x')).  1 − µ(y)  1 − µ(y) 1 − µ(y') = .∴ log 2   1-µ(x) 1 − µ(x')  1-µ(x)  Now .  1 − µ(y')  = log 2    1 − µ(x')  ∴The map is well defined. Again,  1 − µ(yy ')  (x/y)U.(x'/y')U=(xx'/yy')U → log 2  = 1 − µ(xx ')   1 − µ(y) − µ(y ') + µ(y)µ(y ')  log 2   1 − µ(x) − µ(x ') + µ(x)µ(x ') 

(1 − µ(y) )(1 − µ(y ') ) = (1 − µ(x) )(1 − µ(x ') )  1 − µ(y)   1 − µ(y ')  log 2   + log 2   1 − µ(x)    1 − µ(x ') 

Therefore the map is a homomorphism. The map is 1-1, for,

 1 − µ(y)   1 − µ(y')  log 2   = log 2    1-µ(x)   1 − µ(x')  ⇒ 1-µ ( x' ) -µ(y)+µ(y)µ(x')= 1-µ ( y' ) -µ(x)+µ(x)µ(y')

⇒ µ ( yx' ) =µ ( xy' ) ⇒ ( yx' ) =

( xy' ) ⇒

yx'=uxy', u is a unit in V

⇒ x'/y' = u x/y ⇒ (x'/y')U = (x/y)U. Also the map is on to. Hence G is isomorphic to P. ∎ 4.2. Definition. Let V be an integral domain and µ be an ℝ-valuation fuzzy ideal of V. Then the group P= is called the value group of µ. In the following proposition, we prove that for any given subgroup G of ℝ, we can construct an ℝ-valuation fuzzy ideal with value group G. We require the following theorem.

4.3. Theorem[18]. Let R be an integral domain and G an ordered abelian group. If there exists a map υ0 from R into G satisfying the conditions of a valuation, then there exists a valuation υ on its quotient field K with value group G and valuation ring V such that υ(a) = υ0(a) for all a ∈ R and R is a subring of V. ∎

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

4.4. Proposition. Given an ordered subgroup G ⊆ ℝ, there exist an integral domain R and an ℝ-valuation fuzzy ideal µ on R having value group G. Proof. Let G be a subgroup of ℝ. Let k be a field say k = Z2, the binary field. Let R be the vector space over k with basis { xa : a ∈ G, a ≥ 0 }. Define multiplication of basis elements by xa.xb = xa+b and extend this to a product operation on R by linearity. Then R is an integral domain, the unit element being x0 = 1[1]. Define the map υ0 : R ⟶ G by υ0(0) = and

υ0 (x a1 +.......+x a k ) = min{a1 ,....,a k } a

If X = x a1 +.....+ x j and Y = x b1 +.....+ x bk are elements of R, then υ0(X + Y) ≥ υ0(X) ∧ υ0(Y) and υ0(XY) = υ0(X) + υ0(Y). By theorem 4.3, the extension υ of υ0 to the quotient field K of R defined by υ(X/Y) = υ0(X) - υ0(Y) is a valuation on K. The valuation ring being V = {X/Y : X, Y ∈ R; υ(X/Y) ≥ 0}. The fuzzy ideal µ of V defined by µ(X/Y) = 1 − 2−υ (X/Y) is an ℝ-valuation fuzzy ideal of V with value group G. In fact,  1-µ(X1 /Y1 )    log2   : 0 ≠ X1 / Y1 ∈ V, 0 ≠ X2 / Y2 ∈ V 1-µ(X /Y )    2 2   

= {υ( X2 /Y2 ) − υ( X1 / Y1 ) : 0≠ X1 /Y1 ∈V, 0 ≠ X2 / Y2 ∈V} = G. ∎ 5. Construction of an ℝ- valuation fuzzy ideal in a discrete Valuation ring V Since discrete valuation rings are real valuation rings, they always possesses ℝ-valuation fuzzy ideals. Instead of generating ℝ-valuation fuzzy ideals from

valuations, we here develop a method for finding ℝ-valuation fuzzy ideals by assigning a value to its single irreducible element. An ℝ-valuation fuzzy ideal µ essentially satisfy µ * = {0} and µ(x) + µ(y) = µ(xy) + µ(x) µ(y) for all x, y. It follows from the proof of proposition 3.4 that µ(x) = 0 ⟺ x is a unit. Again, if y ∈ V, µ(0) + µ(y) = µ(0) + µ(0) . µ(y). ∴µ(y) [µ(0) – 1] = 0. Since y is arbitrary, µ (0) - 1= 0, hence µ(0) = 1. The ideal structure of a DVR is R ⊃ M = (a) ⊃ (a2) ⊃ (a3) ⊃………⊃ (0) where ‘a’ is the single irreducible element of V. Now, µ(a) + µ(a) = µ(a2) + µ(a). µ(a). ∴µ(a2) = µ(a) (1-µ(a)) + µ(a). Again, µ(a2) + µ(a) = µ(a3) + µ(a2). µ(a). ∴µ(a3) = µ(a2) (1-µ(a)) + µ(a). Similarly, µ(a4) = µ(a3) (1-µ(a)) + µ(a) and µ(a5) = µ(a4) (1-µ(a)) + µ(a). In general, µ(ak) = µ(ak-1) (1-µ(a)) + µ(a), k = 1,2,3….. (1) Thus to get an ℝ-valuation fuzzy ideal, fix a membership value for ‘a’. It must be less than 1, for, otherwise µ(x) = 1 ∀ x ∈ M so that µ * ≠ 0 which is not possible. Then use the above recurrence formula (1) to find the values of µ(a2), µ(a3), µ(a4) etc. Now define µ by

µ(x) = 0, if x ∈ V – (a) = µ(ak), if x ∈ (ak) – (ak+1), k = 1, 2, ……. = 1, if x = 0. To prove that µ is an ℝ-valuation fuzzy Ideal, note that from the definition of µ itself, µ∗ = {0}. In order to prove that µ(x) + µ(y) = µ(xy) + µ(x)µ(y), we first prove the following.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

(i) µ(ah) – µ(ak) = µ(ah-k)(1 - µ(a))k ; h,k = 1, 2, 3, …… , h ≥ k (ii) µ(ak) = µ(a)[k – (µ(a) + µ(a2) + …….. + µ(ak-1 ))], k = 2, 3, 4, ……. (iii) µ(ak) = µ(a)[1 + t + t2 + ……… + tk-1], where t = 1 - µ(a), k = 2, 3, 4, ……. From the recurrence formula (1), µ(ah) = µ(ah-1) (1- µ(a)) + µ(a) and µ(ak) = µ(ak-1) (1- µ(a)) + µ(a). Hence, µ(ah) – µ(ak) = (1 – µ(a)) (µ(ah-1) – µ(ak-1)) = (1 – µ(a))2 (µ(ah-2) – µ(ak-2)). Proceeding like this, we get µ(ah) – µ(ak) =(1 – µ(a))k-1 (µ(ah-k+1) – µ(a)) =(1–µ(a))k-1 µ(ah-k)(1 - µ(a)) using(1) =(1–µ(a))k µ(ah-k). This proves (i). Again, from the recurrence relation (1), µ (a) = 0 (1–µ(a)) + µ(a) and µ (ai+1)= µ(ai) (1–µ(a)) + µ(a), i = 1, 2, ….., k-1. Adding these k equations, we get µ(ak) = - µ(a)[ µ(a) + µ(a2) + ….. + µ(ak-1) ] + k µ(a) = µ(a)[ k – (µ(a) + µ(a2) + ….. + µ(ak-1)) ]. This proves (ii). Finally, µ(a2) = µ(a) (1 – µ(a)) + µ(a) = µ(a)[ 1 + (1 – µ(a)) ] = µ(a) [ 1 + t ], where t = 1 – µ(a). Hence, µ(a3) = µ(a2) (1 – µ(a)) + µ(a) = µ(a) [ 1 + t ] t + µ(a) = µ(a) [ 1 + t + t2 ]. Therefore by induction, µ(ak) = µ(a) [ 1 + t + t2 + …… + tk-1 ]. This proves (iii). Now if i ≥ j, we have

µ(ai) + µ(aj)- µ(ai+j)= µ(a){ i – [µ(a) + ……. + µ(ai-1)]} + µ(a){ j – [µ(a) + …… + µ(aj-1) ]} – µ(a){ (i + j) - [µ(a) + ……. + µ(ai+j-1)]} (using (iii)) = µ(a){ [µ(ai) + ……. + µ(ai+j-1)] – [µ(a) + ……. + µ(aj-1)]}

= µ(a){ µ(ai) +[ µ(ai+1) - µ(ai)] +……. + [µ(ai+j-1) - µ(aj-1)]} = µ(a){ µ(ai) + µ(ai)[1- µ(a)] +……. + µ(ai)[1 - µ(a)]j-1} (using (i)) i i = µ(a){µ(a ) + µ(a ) t + ……… +µ(ai) tj-1} = µ(ai) µ(a) {1 + t + …….. + tj-1} = µ(ai) µ(aj) using (ii). Hence, µ(ai) + µ(aj) = µ(ai+j) + µ(ai) µ(aj) (2) We now prove that µ(x) + µ(y) = µ(xy) + µ(x) µ(y), for all x,y ∈ V. (i) If x ,y ∈ V – (a), then x and y are units. ∴ xy is a unit, hence belongs to V - (a). By definition of µ, µ(x) = µ(y) = µ(xy) = 0. ∴ µ(x) + µ(y) = µ(xy) + µ(x) µ(y). (ii) If x, y ∈ (ai) – (ai+1) and i ≥ 1, then x = rai , y = sai where r and s are not multiples of a. ∴ xy = rs a2i ∈ (a2i) – (a2i+1). Now µ(x) = µ(y) = µ(ai) and µ(xy) = µ(a2i). By (2), µ(x) + µ(y) = µ(ai) + µ(ai) = µ(a2i) + µ(ai) µ(ai) = µ(xy) + µ(x) µ(y). (iii) If x ∈ V – (a) and y ∈ (aj) – (aj+1), j ≥ 1, then x is a unit, hence xy ∈ (aj) – (aj+1). ∴ µ(x) = 0 and µ(y) = µ(xy) = µ(aj). ∴ µ(x) + µ(y) = µ(xy) + µ(x) µ(y). (iv) If x ∈ (ai) – (ai+1) and y ∈ (aj) – (aj+1) and i > j, then x = rai and y = saj where r and s are not multiples of a. ∴ xy = rs ai+j ∈ (ai+j) – (ai+j-1). Now µ(x) = µ(ai), µ(y) = µ(aj) and µ(xy) = µ(ai+j). By (2), µ(x) + µ(y) = µ(ai) + µ(aj) = µ(ai+j) + µ(ai) µ(aj) = µ(xy) + µ(x) µ(y). Finally, to prove that µ(y) ≤ µ(x) ⟹ x/y ∈ V, let x, y ∈ V. If µ(y) < µ(x), then there exist i and j with i < j such that y ∈ (ai) – (ai+1) and x ∈ (aj) – (aj+1). Now y = rai and x = saj where r and s are not multiples of a, hence units. ∴ x/y = (s/r) aj-i ∈ V. If µ(y) = µ(x), then x, y ∈ (ai) – (ai+1) for some i. ∴ y = rai and x = sai where r and s are units, hence x/y = s/r ∈ V. Thus µ is an ℝvaluation fuzzy ideal. ∎

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

5.1. Example. Consider the localization Z(2) of the ring of integers Z. Then V = Z(2) ⊃ (2/1) ⊃ (22/1) ⊃ (23/1) ⊃…………⊃ (0) is a discrete valuation ring. Take µ(2/1) = 1/2, µ(22/1) = (1/2)(1- 1/2) + 1/2 = 3/4, µ(23/1) = 3/4(1- 1/2) + 1/2 = 7/8. etc. Define µ by µ(x) = 0, if x ∈ Z(2) – (2/1) = (2i – 1)/2i, if x ∈ (2i/1) – (2i+1/1), i = 1, 2, ….. and µ(0) = 1. Then µ is an ℝ-valuation fuzzy ideal on V. ∎

5.2. Remark. Arbitrarily fixing a value for µ(a) as in example 5.1, we can define infinitely many ℝ-valuation fuzzy ideals in a DVR. Equivalently, we can define infinitely many valuations. ∎ REFERENCES 1. David Eisenbud, Commutative Algebra with a view Toward Algebraic Geometry, Springer-Verlag, New York, (1995). 2. N S Gopalakrishnan, Commutative Algebra, Vikas Publishing House, New Delhi, (1984). 3. W.J. Liu, Fuzzy Invariant subgroups and Fuzzy Ideals, Fuzzy sets and systems 8, 133-139 (1982). 4. W.J. Liu, Operations on Fuzzy Ideals, Fuzzy Sets and Systems 11(1983) 31-41. 5. J. N. Mordeson and D. S. Malik, Fuzzy commutative Algebra, World Scientific, Singapore, (1998). 6. A. Rosenfeld, Fuzzy Groups, J. Math. Anal. & Appl. 35, 512-517 (1971). 7. Souriar Sebastian and S. Babusundar, On the chains of level subgroups of homomorphic images and pre-images of fuzzy groups, Banyan Mathematical Journal, 1, 25-34 (1994).

8. Souriar Sebastian and S. Babusundar, Existence of fuzzy subgroups of all level cardinality up to ℵ0, Fuzzy sets and systems, 67, 365-368 (1994). 9. Souriar Sebastian and S. Babusundar, Commutative L-fuzzy subgroups, Fuzzy sets and systems, 68, 115-121 (1994). 10. Souriar Sebastian and S. Babusundar, Generalizations of some results of Das, Fuzzy sets and systems, 71, 251-253 (1995). 11. Souriar Sebastian and Thampy Abraham, Fuzzification of Cayley’s and Lagrange’s Theorems, J. Comp. & Math. Sci., 1, 41-46 (2009). 12. Souriar Sebastian and Sasi Gopalan, Approximation studies on image enhancement using fuzzy techniques, International Journal of Advanced Science and Technology, 10, 11-26 (2009). 13. Souriar Sebastian and Sasi Gopalan, Modified fuzzy basis function and function approximation, J. Comp. & Math. Sci., 2, 263-273 (2010). 14. 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). 15. Souriar Sebastian and George Mathew, On a certain class of fuzzy ideals leading to discrete valuation rings, J. Comp. & Mathe. Sci., 1, 690-695 (2010). 16. Souriar Sebastian and George Mathew, On valuation fuzzy ideals of Rings, J. Ultra Scientist of Physical Sci., 22, 833 – 838 (2010).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)

Souriar Sebastian, et al., J. Comp. & Math. Sci. Vol.2 (1), 51-60 (2011)

17. L. A. Zadeh, Fuzzy Sets, Inform and Control 8, 338-353 (1965).

18. O. Zariski and P Samuel, Commutative Algebra (Vols 1 and 2) Springer-Verlag, New York (1979).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 1, 28 February, 2011 Pages (1-169)