Journal of Communication and Computer 13 (2016) 42-49 doi:10.17265/1548-7709/2016.01.007 D DAVID PUBLISHING

On Properties of Uniformly Strongly Fuzzy Ideals

Flaulles B. Bergamaschi1, Alexsandra Oliveira Andrade1 and Regivan H. N. Santiago2 1. Department of Science and Technology, Southwest Bahia State University,Vitoria da Conquista, Bahia 45.031-900, Brazil 2. Department of Informatics and Applied Mathematics, Federal University of Rio Grande do Norte, Natal 59.078-900, Rio Grande do Norte, Brazil

Abstract: The main purpose of this paper is to continue the study of uniform strong primeness on fuzzy setting. A pure fuzzy notion of this structure allows us to develop specific fuzzy results on USP (uniformly strongly prime) ideals over commutative and noncommutative rings. Besides, the differences between crisp and fuzzy setting are investigated. For instance, in crisp setting an I of a R is a USP ideal if the quotient R/I is a USP ring. Nevertheless, when working over fuzzy setting this is no longer valid. This paper shows new results on USP fuzzy ideals and proves that the concept of uniform strong primeness is compatible with a-cuts. Also, the Zadeh’s extension under epimorphisms does not preserve USP ideals. Finally, the t- and m- systems are introduced in a fuzzy setting and their relations with fuzzy prime and uniformly strongly prime ideals are investigated.

Key words: Fuzzy ideal, fuzzy set, uniformly strongly prime, prime.

1. Introduction perfectly possible to define prime rings and prime ideals. This paper begins with the following question: Is it A ring is a set endowed with two operations, possible to have a fuzzy version of a USP (uniformly addition and multiplication, where these operations strongly prime) ideal? have properties to those operations defined for the In 2014, Bergamaschi and Santiago answered this . Prime ideals are subsets of a ring with question by introducing in Ref. [1] the notion of USP similar properties of prime numbers and they develop fuzzy ideals without a-cut dependence for the first an important role in the study of rings as well as prime time. Since then we have been dedicated to investigate numbers in the ring of integers. its properties and similarities with crisp setting. So Prime rings are a special class of rings where the this paper increases our understanding on USP ideals zero ideal (0) is a . In a and fills the gap related with a-cuts as well as it theory prime rings are well-known as an integral introduces the concept of systems for a fuzzy i.e. rings where ab = 0 implies a = 0 or b = 0. environment. The concept of SP (strongly prime) ring appeared in The following paragraphs introduce the whole 1973 made by Lawrence in his Master’s thesis as a subject. subclass of prime rings. In 1975, Lawrence and The importance of prime numbers for pure Handelman [2] presented an extensive work on SP mathematics results and many applications is rings. They developed properties and proved well-known. important results, e.g., all prime rings may be For example, nowadays prime numbers are the embedded in an SP ring; all SP rings are nonsingular. fundamental idea behind cryptography. We may think Besides, they defined the USP ring as a subclass of SP of primeness for sets and subsets. In other words, it is rings and demonstrated some initial uniform Corresponding author: Flaulles B. Bergamaschi, Ph.D., research : fuzzy prime ideals. If the reader has interest in applications about prime ideals and **This research was partially supported by the Brazilian fuzzy prime ideals in coding theory it is worth reading the Research Council (CNPq) under the process 306876/2012-4. introduction of Ref. [14] and its references.

On Properties of Uniformly Strongly Fuzzy Ideals 43 properties. because the crisp definition of USP ideal is more After a little time, the study of USP rings became suitable to be translating to fuzzy environment. This more important. Thus, in 1987, Olson [3] presented a approach became more interesting, since it allowed us paper about USP rings and USP radical. Olson proved to find pure fuzzy results. For instance, it was proved that USP rings generate a radical class which properly that every USPf ideal is a prime ideal according to the contains both the right and left SP radicals and which newest is independent of the famous Jacobson and Brown- Pure fuzzy definition of fuzzy prime ideal was McCoy radicals. Also, some results in theory given by Navarro [14] in 2012. Also, it was proved were rediscovered by using the SP and USP ideals. that some results in fuzzy setting are not true like in In 1965, Zadeh [4] introduced fuzzy sets and in classical theory. Besides, the behavior of Zadeh’s 1971, Rosenfeld [5] introduced fuzzy sets in the realm extension on USPf ideals was studied and as a of group theory and formulated the concept of fuzzy consequence we built a version of correspondence subgroups of a group. Since then, many researchers theorem for USPf ideals. have been engaged in extending the concepts/results This paper expands the last three papers [1, 13] with of to the broader framework of the new properties of USPf ideals and introduces the t- fuzzy setting. Thus, in 1982, Liu [6] defined and and m-systems for a fuzzy setting. It is shown that studied fuzzy as well as fuzzy ideals. every fuzzy ideal is contained in a USPf ideal. Also, it Subsequently, Liu himself [7], Mukherjee and Sen [8], is proved that the complement of prime/USP fuzzy Swamy and Swamy [9], and Zhang Yue [10], among ideal is an m/t-system. Moreover the concept of others, fuzzified certain standard concepts/results on m-systems is compatible with the newest (2012) rings and ideals. For example: Mukherjee was the first definition of fuzzy prime ideals over noncommutative to study the notion of prime ideal in a fuzzy setting. rings given by Navarro et al. in Ref [14]. It is also Those studies were further carried out by Kumar in shown that the inverse image of a USPf ideal is a Refs. [11, 12], where the notion of nil radical and USPf ideal when the mapping is a ring semiprimeness were introduced. homomorphism extended by Zadeh’s principle. In 2013, motivated by crisp problems in group ring However, on the direct image this is no longer valid. theory (e.g. isomorphism problem) we proposed in Finally, the paper leaves for the reader some open Ref. [13] the concept of SP ideal for fuzzy questions about USP investigation on fuzzy setting. environment for the first time. The main goal was to This paper has the following structure: Section 2, investigate this structure in fuzzy environment. Thus, which not only provides an overview about the ring the concept of SP fuzzy ideal was born and it was and fuzzy ring theory, but it also contains the classical called SPf ideal. The difficulty to find a pure fuzzy definition and results of USP rings/ideals; Section 3 definition of an SP ideal forced us to define SPf ideals has the definition of USPf ideals and the compatibility on cuts. But this approach had some issues. One of with cuts; Section 4 shows that Zadeh’s extension them lies in the fact that all results have similarities in under epimorphisms does not preserve uniform strong crisp algebra. In other words, fuzzy setting became a primeness; Section 5 deals with t-/m-systems and its mirror of classical theory. relations with primeness and uniform strong Although a pure definition of SPf ideals was not primeness; Section 6 provides the final remarks. founded, the ideas developed in Ref. [13] enabled us 2. Preliminaries to create the definition of USPf (uniformly strongly prime fuzzy) ideal. The concept of USPf was possible This section explains some definitions and results

44 On Properties of Uniformly Strongly Fuzzy Ideals

that will be required in the next sections. All rings are injn: ik∈ I and jk ∈ J, k = 1,...,n; where n ∈ associative with identity and are usually denoted by R. Z+}and the set x Ry = { x r y : r ∈ R}. Here, we expose primitive definitions of prime Definition 3 A prime ideal in an arbitrary ring R is rings/ideals and Uniformly Strongly Prime any proper (P  R and P  R) ideal P such that, rings/ideals. whenever I, J are ideals of R with IJ  P, either I  P Definition 1 A ring is a nonempty set R of elements or J  P. closed under two binary operations + and • with the Proposition 1 ([15], Proposition 10.2) An ideal P following properties: of a ring R is prime if for x, y∈ R, xRy  P implies x (R, +) (that is, the set R considered with the single ∈ P or y ∈ P. operation of addition) is an abelian group (whose Definition 4 An ideal P of a ring R is called identity element is denoted 0R, or just 0); completely prime if given a and b two elements of R The operation • is associative: (a•b)•c = a• (b•c) for such that their product ab∈ P, then a∈ P or b∈ P. every a, b, c e R. Thus, (R, •) is a semigroup; Given a ring R and a ∈ R, the set (a) = RaR =

The operations + and • satisfy the two distribu- tive {x1ay1 + … + xnayn： n ∈ N, xi,yi∈ R} is an ideal laws: (a+b) •c = a•c+b•c and a• (b+c) = a•b+a•c, for and is called the ideal generated by a. every a, b, c∈ R. Definition 5 Let A be a subset of a ring R. The right

If R is a ring and there exists an element 1 such that annihilator of A is defined as Anr (A) = {x ∈R: Ax a • 1 = a for every a e R we say that the ring has = (0)}. Similarly, we can define the left annihilator of multiplicative identity. Also, if a • b = b •a for a, b ∈ A. R we call R a commutative ring. Definition 6 [2] A ring R is called right strongly Very often we omit writing the • for multiplication, prime if for each nonzero x∈R there exists a finite that is, we write ab to mean a • b. Note that there can nonempty subset Fx of R such that the A n r (xFx )= only be one additive identity in R (because (R, +) is a (0). group, and a group can only have one additive When R is right strongly prime we can prove that Fx identity). Also, there can be only one multiplicative is unique and called a right insulator for x. identity in R. If R is commutative and for any a, b∈ Handelman and Lawrence worked exclusively with R, ab = 0 implies a = 0 or b = 0 we call R an integral rings with multiplicative identity. However, Parmenter, domain. Note that the ring of n × n matrices with Stewart and Wiegandt [16] have shown that it is integers entries is a and nor an equivalent to: . Definition 7 A ring R is right strongly prime if each Definition 2 Let R be a ring. A nonempty subset I nonzero ideal I of R contains a finite subset F which of R is called a right ideal of R if: has right annihilator zero. (a) a,b∈ I implies a + b ∈ I; It is clear that every right strongly prime ring is a (b) given r ∈ Ra ∈ I, then ar∈ I (that is, a right prime ring. It is also possible to define left strongly ideal absorbs right multiplication by the elements of prime in a manner analogous to that for right strong the ring). primeness.Handelman and Lawrence showed that Similarly we can define left ideal replacing (b) by: these two concepts are distinct, by building a ring that (b´) given r∈R, a∈I, then ra∈I. If I is both right and is right strongly prime but not left strongly prime ([2], left ideal of R, we call I a two-sided ideal or simply an Example 1). ideal. Example 1 Consider Zn the commutative ring of

For the next definition consider: IJ = {i1j1 +…+ integers mod n, for n > 1. If a ∈ Z, the class of a is

On Properties of Uniformly Strongly Fuzzy Ideals 45

[a] = {x ∈Z: (x mod n) = a}. Note that if n is not a nonzero elements x and y of R/I (the complement of I , then there exists p, q ∈ Z such that n in R), there exists f ∈ F such that xfy I. = p q, where 0< p < n and 0< q < n. Hence, [ p q ] =0 Proposition 2 An ideal I of a ring R is a USP ideal in Zn, but [ p ]  0 and [ q ]  0. We conclude that if the quotiente R/I is a USP ring.

Zn is not a integral domain and as a consequence Zn is Proposition 3 An ideal I of a ring R is a USP if not a prime ring. Thus, Zn is not right strongly prime there exists a finite set F  R such that xFy  I implies ring. On the other hand, if n is prime, Zn is a field, x ∈ I or y ∈ I, where x, y∈ R. hence right strongly prime ring. From this point forward consider  as infimum Definition 8 A ring is a bounded right strongly and  as supremum in [0, 1]. prime ring of bound n, if each nonzero element has an Definition 11 A fuzzy subset I: R [0, 1] of a ring insulator containing no more than n elements and at R is called a fuzzy of R if for all x, y∈ R the least one element has no insulator with fewer than n following requirements are met: elements. (1) I(x - y)  I(x)  I(y); Definition 9 A ring is called uniformly right (2) I ( x y )  I(x)  I(y); strongly prime if the same right insulator may be If condition (2) is replaced by chosen for each nonzero element. I ( x y )  I (x)  I ( y ) , then I is called a Since an insulator must be finite, it is clear that fuzzy ideal of R. every uniformly strongly prime ring is a bounded right Note that using properties of the t-norm we have for strongly prime ring of bound n. Again, analogous any fuzzy subring/fuzzy ideal I of a ring R the definitions of bounded left strongly prime and following: if for some x, y∈ R, I(x) < I(y), then I(x uniformly left strongly prime can be formulated. As — y) = I(x) = I(y — x); Also, if I is a fuzzy ideal of a was the case with the notation of strong primeness it is ring R, then I(1)  I(x)  I(0) for all x∈ R. possible to find rings which are bounded left strongly Given a fuzzy set I and a in [0, 1] the set Ia ={x∈R; prime but not bounded right strongly prime, and such that I(x) a} is called a-cut of I. vice-versa (see Ref. [2], Example 1). However, Olson Proposition 4 [4] A fuzzy subset I of a ring R is a

[3] showed that the concept of uniformly strongly fuzzy ideal of R if all a-cuts Ia are ideals of R. prime ring is two-sided in view of the following Definition 12 (Zadeh’s Extension) Let f be a result: function from set X into Y, and let  be a fuzzy Lemma 1 [ 3 ] A ring R is right/left uniformly subset of X. Define the fuzzy subset f (  ) of Y in the strongly prime if there exists a finite subset F  R following way: For all y∈ Y, such that for any two nonzero elements x and y of R, there exists f ∈ F such that xfy  0. Corollary 2 [3] R is uniformly right strongly prime ring if and only if R is uniformly left strongly prime If  is a fuzzy subset of Y, we define the fuzzy ring. subset of X by f-1(  ) where Lemma 3 [3] The following are equivalent: f-1(  )(x) =  (f (x)). (i) R is a uniformly strongly prime ring; Definition 13 [14] Let R be a ring with unity. A (ii) There exists a finite subset F  R such that xFy nonconstant fuzzy ideal P: R [0, 1] is said to be =0 implies x = 0 or y = 0, where x, y∈R; prime or fuzzy prime ideal if for any x, y∈ R,  Definition 10 An ideal I of a ring R is called USP P(xRy)=P(x)  P(y). ideal if there exists a finite set F  R such that for two Proposition5 [14] Let R be an arbitrary ring with

46 On Properties of Uniformly Strongly Fuzzy Ideals unity and P: R [0, 1] be a nonconstant fuzzy ideal finite set given by definition 14. Let x, y ∈ R and of R. The following conditions are equivalent: I(1) < a < I(0) such that xFy  Ia. Hence, I(x)  (i) P is prime; I(y)=  I(xFy)  a, and thus I(x)  a or I(y)  a.

(ii) P a is prime for all P ( 1) < a  P(0); Therefore, x ∈Ia or y ∈ Ia. On the other hand,

(iii) R / P a is a prime ring for all P ( 1 ) < a  P(0); suppose Ia is a USP ideal of R for all I(1) < a  I(0).

(iv) For any fuzzy ideal J, if J ( x r y )  P(xry) According to proposition 3 each Ia has a finiteset Fa for all r ∈ R, then J ( x )  P(x) or J(y)  such that if xFay  Ia implies x∈Ia or y ∈ Ia. Let a P(y). finite set F =  Fa. Suppose I (1) I(x)  I(y) and t =  I(xFy) for any x,y ∈ define fuzzy left cosetr +1, where (r +1)(x) = I(x — r). R. Note that t > I (x)  I (y) and t  I (xfy) for all

Given an ideal (crisp or fuzzy) of R define R/I = {r f ∈ F. Hence, x,y It, but xFy  It and thus (by

+1: r e R} the quotient ring by I. In R/I we can define hypothesis) x ∈ It or y ∈ It, where we have a + and •,where (r +I)+（ s +I) = (r +s)+I and (r + I) • contradiction. Therefore,  I(xFy) = I(x)  I(y). (s + I) = (rs) + I. Corollary 4 If I is a USPf ideal of a ring R if R/I is Proposition 6 [3] If I and P are ideals of a ring R a USP ring for all I (1) < a  I (0). with P a USP ideal, then I  P is a USP ideal. Corollary5 If I is USPf ideal, then I is fuzzy prime ideal. 3. Uniform Strong Primeness For the following result Ker(f) = {x ∈ R : f (x)= 0} This section introduces the concept of Uniformly is the kernel of the homomorphism f and f-1(J) is the Strongly Prime fuzzy ideal (or shortly USPf ideal) fuzzy subset of R by Zadeh’s extension. according to definition given in Ref. [1]. Proposition9 [1]If f:R  S is a homomorphism Definition 14 [1] Let R be an associative ring with of rings and J USPf ideal of S, then f-1(J) is USPf ideal unity. A non-constant fuzzy ideal I: R [0, 1] is said of R which is constant on Ker(f). to be Uniformly Strongly Prime fuzzy ( USPf) ideal if After proposition 9 we can think about the direct there exists a finite subset F such that  I( x F y ) = image. In other words, if I is a USPf ideal of R which I(x)  I(y), for any x,y∈ R. The set F is called is constant on Ker(f), then f (I) is USPf is an ideal? In insulator of I. this paper we proved this statement as false, according Proposition 7 Let R be an associative ring with to proposition 13 in the next section. unity and Aa crisp USP ideal of R. Then I, where For the next result consider I* = II(0 ) = {x ∈ R: I(x) I(x)=1 if x in A, and I(x)=0 otherwise, is a USPf ideal = I(0)}. of R. Proposition 10 [1] If I is a USPf ideal of a ring R,

Proof Clearly I is a fuzzy ideal. According to then R/I*  R/I. definition 10 given x, y ∈ R there exists a finite set Corollary 6 [1] If f :R  S is an epimorphism F  R such that if xFy  A implies x ∈ A or y∈A. and I USPf ideal of R which is constant on Ker(f), Suppose xFy  A, then  I(xFy) = 1 = I(x)  I(y). then R/I  S/f (I). If xFy A, then there exists f ∈ F such that xfy Proposition 11 [1] Let J be a crisp ideal of R. A. Hence, x, f, y  A. Therefore,  I(xFy) = 0 = I(x) Define I: R  [0,1] as I(x)=1 if x=0, I(x)=a if  I(y). x∈J\{0}, I(x)=0 if x J, where a∈(0,1).

Proposition 8 I is USPf ideal of R if Ia is USP ideal Then: of R for all I(1) < a < I(0). (i) I is a fuzzy ideal; Proof Suppose Ia USPf ideal and let F  R be a (ii) I is a USPf ideal if J is USP ideal.

On Properties of Uniformly Strongly Fuzzy Ideals 47

Corollary 7 Let I be a nonconstant fuzzy ideal of R, Consider the ideal I * = {x ∈R :I (x) > I (1)}, by and define M(x)=I (0) if x=0, M(x)=if x∈I*\{0}, Zorn’s Lemma, there exists a maximal ideal M of R

M(x)=I(1) if xI*.Then, M is a USPf ideal of R if I* is containing I*. Now we can define the following fuzzy a USP ideal of R. set: K(x) =I(0) if x ∈ M, K(x)=I(1) if xM. Clearly, Corollary 8 Let I a fuzzy ideal of a ring R and Im(I) K is a fuzzy ideal and I  K. Now, consider the is a finite set. Then, I is a USPf ideal if Ia is a USP finite set F = {1}. Thus,  K(xFy)= K(xy) for any x, y ideal for all I(1) < a  I(0). ∈ R. If x ∈ M or y ∈ M, then xy∈ M and then Example 2 Consider Z the ring of integers and 4Z = K(xy) = I(0) = K(x)  K(y). On the other hand, as R {x ∈Z: x = 4q, q ∈ Z}. Define I(x)=1 if x=0, is commutative, M is completely prime, hence if x  I(x)=1/2 if x∈4Z\{0}, I(x)=0 if x4Z. It is easy to M and y  M, then xy M. Therefore, K(xy) = I(1) see that I is a f u z z y ideal, since its all a-cuts (I1 = (0), = I(1)  I(1) = K(x)  K(y).

I1/2 = 4Z, I0 = Z) are ideals. Moreover, I is not USPf Conjecture 1 According to definition of fuzzy ideal, since 4Z is not prime ideal, acccording to maximal ideal given by Malikin [17] , Kin the proposition 11. Note that I* = (0) is USP ideal. Hence, demonstration of proposition 12 is a fuzzy maximal

R/I*is a USP ring. Applying the proposition 10R/I  ideal.

R/I*. Therefore, R/I is a USP ring, although I is not a Conjecture 2 The propostion 12 can be extended to USPf ideal. noncommutative rings. The next example shows us that direct image of Proposition 13 Let f: R S is a epimorphism of USPf ideals by Zadeh’s extension on homomorphism commutative and not USP rings. If I is a USPf ideal of are not USPf ideals. R which is constant on Ker(f), then f (I) is not a

Example 3 [1] Let f: Z Z4 be defined by f (x) = USPf ideal of S. Proof As I is constant on Ker(f), then by x 4 = x mod 4. Then, the function f is an Proposition 10 and Corollary 6 we have: R/I*  R/I epimorphism with kernel 4Z. Consider I(x)=1 if x=0,  R/f (I)  R/f (I) *. As I* is USP ideal, then R/I*

I(x)=1/2 if x∈3Z\{0}, I(x)=0 if x3Z. Also consider is USP ring. Hence, R/f (I)* is USP ring. Thus, f (I)* is f(I)(y)=1 if x = 0, f(I)(y)=1/2 if x  0. Clearly I is USP ideal. As we know f (I)*  f (I)a for all a ∈

USPf ideal of Z, but f (I) is not a USPf ideal of Z4, [0,1]. But S is commutative and f (I)* is Prime, hence f since I1/2 = Z4 is not a USP ideal. (I)* is maximal, this last statement implies f (I)a = S for all a  I(0) and by hypoteses S is not USP ring. 4. Uniform Strong Primeness under Therefore, f (I) is not a USPf ideal. Homomorphism Question 1 The proposition 13 shows us that USPf This section amplifies results about USPf ideals. ideals cannot preserved by Zadeh’s extension. Thus, The first one (proposition 12) is geared to we ask: Under which conditions can Zadeh’s commutative rings. But it may be valid in extension preserves the USPf ideals? noncommutative case (conjecture 2). The proposition Proposition 14 If I and P are fuzzy ideals of a ring 3 shows the difference between crisp and fuzzy setting R with P is a USPf ideal, then I  P is a USPf ideal of by showing the behavior of Zadeh’s extension on R. USPf ideals. Proof Note that: (I  P)(xFy)=( I(xFy))  Proposition 12 If I is a nonconstant fuzzy ideal of a ( P(xFy))=( I(xFy)) （ P(x))  P(y)  P(x) commutative ring R, then there exists a USPf ideal K  P(y). such that I  K. Proposition 15 Any USPf ideal contains properly

48 On Properties of Uniformly Strongly Fuzzy Ideals another USPf ideal. exists r ∈ R such that xry∈ M. 1 Proof Suppose I USPf ideal of a ring R. Let P = I Definition 16 [3] A subset T of a ring R is called a 2 t-system if there exists a finite set F  R such that for 1  I defined by P(x) = I(x) for all x ∈ R. Hence, any two elements x, y∈T there exists f ∈ F such 2 IxFy Ix Iy that xfy∈ T. PxFy Px() Py () 222 It is not hard to prove that F is unique. So, it will be The following proposition tell us about the called the insulator of T. Note that, the empty set will following question: If a fuzzy ideal has at least one be a t-system. USP a-cut, what can we say about this ideal. Is it a Proposition 17 [18] If M is a t-system, then M is a USPf ideal? m-system. Proposition 16 Let I is a nonconstant fuzzy Ideal of Proposition 18 [18] I is a prime ideal of a ring R if a integral domain R and R is not a USP ring and It is R \ I (the complement of I in R) is an m-system.

USP ideal for some I (1) < t  I (0). If k  t and Ik Proposition 19 [3] An ideal I is a USP ideal of a

 It, then Ik is not a USP ideal. Hence, I is not a ring R if R\I (the complement of I in R) is a t-system. USPf ideal. For the following definition consider xRy = {xry:r

Proof When Ik= R is trivial. Now suppose Ik  R ∈R}. and note that in a integral domain if I is a USP ideal, Definition 17 Let R be an associative ring with then I is a Maximal ideal. Thus, It  Ik is impossible, unity. A nonconstant fuzzy set K: R [0,1] is said to since it is Maximal. If Ik  It implies Ik not maximal, be a fuzzy m-system if K(xRy) = K(x)  then Ik not a USP ideal. K ( y ) , for any x , y ∈ R. 5. The Fuzzy m- and t-Systems Proposition 20 If K is a fuzzy subset of a ring R such that Ka is an m-system for all a-cuts, then An m-system is a generalization of multiplicative K(xRy) K (x)  K(y), systems. In the ring theory a set M is an m-system if Proof Let x, y∈ R and t = K(x)  K(y). As Kt is for any two elements x, y in M there exists r in R such an m-system and x, y ∈Kt then there exists r ∈ R that the product xry belongs to M. It is not hard to such that xry∈Kti.e K (xry)  t. Hence, K(x R y) perceive that an ideal is prime if its complement is an t. m-system (Mccoy [18]). On the other hand we have Question 2 Under which conditions may we have the t-systems which are sets where given any two the following result: K is a fuzzy m-system of R if Ka elements x, y in T there exists a finite set F such that is a m-system for all a-cuts? xfy belongs to T for some f in F. Clearly a t-system is For the following results consider P the fuzzy ideal an m-system. Olson [3] proved that I is a uniformly and Pc = 1 –P the complement of P in R. strongly prime ideal if its complement is a t-system. Corollary 9 P is a fuzzy prime ideal of R if Pc is a In this section we will introduce the m-systems in a fuzzy m-system. fuzzy setting based on the definition of fuzzy prime Proof Suppose P fuzzy prime, then ideals defined by Navarro [14] in 2012. The fuzzy t- P (xRy) = P (x) P(y) for any x, y∈ R. Hence, systems are also introduced. As we shall soon see it is  Pc(xRy)= (1- P(x R y))= 1 - P(x R y)= 1 -(P(x) possible to prove that an ideal I is a fuzzy prime ideal P(y)) = (1 - P(x))  (1- P(y)) = Ic(x)  Ic(y). if its complement is an m-system. Moreover, I is USPf Suppose now Pc is a fuzzy m-system, then Pc (x R y) ideal if its complement is a t-system. = Pc(x)  Pc(y) for any x, y∈R. Thus, 1 - P(x R y) Definition 15 [18] A subset M of a ring R is called = 1 - (P(x) P(y)). Therefore P(x R y)= P(x)P(y). an m-system if for any two elements x, y∈ M there For the following definition consider the subset

On Properties of Uniformly Strongly Fuzzy Ideals 49

xFy = {xfy : f∈F}of ring R. University of Rio Grande do Norte) for their financial Definition 18 Let R be an associative ring with support. unity. A nonconstant fuzzy set M: R [0,1] is said to References be a fuzzy t-system if there exists a finite subset F such that M(xFy) = M(x)  M(y)，for any x, y∈ R. [1] Bergamaschi, F. B., and Santiago, R. H. 2014. “A Fuzzy Version of Uniformly Strongly Prime Ideals.” Presented Proposition 21 I is a USPf ideal of R if I (the c at the 2014 IEEE Conference on Norbert Wiener in the complement of I in R) is a fuzzy t-system. 21st Century (21CW). Proof Suppose I USPf, then there exists a finite set [2] Handelman, D., and Lawrence, J. 1975. “Strongly Prime F where I (xFy) = I(x) I(y) for any x, y∈R. Rings.” Transactions of the American Mathematical Society 211: 209-23. Hence,I (xFy) =(1 — I(xFy)) = 1 - I(x F y) = 1 - c [3] Olson, D. 1987. “A Uniformly Strongly Prime Radical.” (I(x) I(y)) = (1 -I(x))  (1 -I(y)) = Ic(x)  Ic(y). J. Austral. Math. Soc. (Series A) 43: 95-102. Suppose now Ic is a fuzzy t-system, then there exists a [4] Zadeh, L. 1965. “Fuzzy Sets.” Information and Control 8: 338-53. finite set F where Ic (xFy) = Ic(x) Ic(y) for any x,y [5] Rosenfeld, A. 1971. “Fuzzy Groups.” Journal of ∈ R. Thus, 1 -I(xFy) = 1 - (I(x) I(y)). Therefore Mathematical Analysis and Applications 35: 512-17. I(x Fy) = I(x) I(y). [6] Liu, W. J. 1982. “Fuzzy Invariant Subgroups and Fuzzy Proposition 22 If M is a fuzzy t-system of R, then Ideals.” Fuzzy Sets and Systems 8: 133-9. [7] Jin, L. W. 1983. “Operations on Fuzzy Ideals.” Fuzzy Ma is a t-system for all a-cuts. Sets and Systems 11: 19-29. Proof As M is a fuzzy t-system there exists a finite [8] Mukherjee, T., and Sen, M. 1987. “On Fuzzy Ideals of a set F, where I(x F y) = I(x) I(y) for any x, y∈ R. Ring.” Fuzzy Sets and Systems 21: 99-104. Let x, y∈ Ma, then  M(x F y) = M(x)  M(y) a. [9] Swamy, U., and Swamy, K. 1988. “Fuzzy Prime Ideals of Since F is a finite set, there exists f∈F such that M Rings.” Journal of Mathematical Analysis and Applications 134: 94-103. (xfy) > a. Thus, xfy ∈ Ma. Therefore, Ma is a t-system. [10] Yue, Z. 1988. “Primel-Fuzzy Ideals and Primary-Fuzzy Question 3 Under which conditions may we have Ideals.” Fuzzy Sets and Systems 27: 345-50. the following result: If T is a fuzzy t-system of R, then [11] Kumar, R. 1991. “Fuzzy Nil Radicals and Fuzzy Primary T is an m-system? Ideals.” Fuzzy Sets and Systems 43: 81-93. [12] Kumar, R. 1991. “Fuzzy Semiprimary Ideals of Rings.” 6. Conclusion Fuzzy Sets and Systems 42: 263-72. [13] Bergamaschi, F. B., and Santiago, R. H. 2013. “Strongly Prime ideals are structural pieces of a ring and Prime Fuzzy Ideals over Noncommutative Rings.” should be the first part in the study of its properties. Presented at the 2013 IEEE International Conference on Fuzzy Systems (FUZZ). As it is known we can decompose an ideal in the [14] Navarro, G., Cortadellas, O., and Lobillo, F. 2012. product of prime ideals. Thus, the following question “Prime Fuzzy Ideals over Noncommutative Rings.” Fuzzy is immediate: may we decompose a fuzzy ideal in a Sets and Systems 199: 108-20. product of fuzzy prime ideals? To answer this [15] Lam, T. Y. 2013. A First Course in Noncommutative Rings. Springer Science & Business Media. question we need to first of all understand primeness [16] Parmenter, M., Passman, D., and Stewart, P. 1984. “The in fuzzy setting. So, this paper contributes to this end Strongly Prime Radical of Crossed and also develops some thoughts on fuzzy ring theory. Products.”Communications in Algebra 12: 1099-113. [17] Malik, D. S., and Mordeson, J. N. 1991. “Fuzzy Maximal, Acknowledgment Radical and Primary Ideals of a Ring.” Information sciences 53: 237-50. The authors would like to thank UESB (Southwest [18] McCoy, N. H. 1949. “Prime Ideal Sin General Rings.” Bahia State University) and UFRN (Federal American Journal of Mathematics: 823-33.