The Strongly Prime Radical of a Fuzzy Ideal
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1 1 The Strongly Prime Radical of a Fuzzy Ideal Flaulles B. Bergamaschi and Regivan H.N. Santiago Departamento de Informática e Matemática Aplicada Universidade Federal do Rio Grande do Norte Natal, Brasil flaulles@yahoo.com.br, regivan@dimap.ufrn.br In 2013, Bergamaschi and Santiago 1 proposed Strongly Prime Fuzzy(SP) ideals for commutative and noncommutative rings with unity, and investigated their properties. This paper goes a step further since it provides the concept of Strongly Prime Radical of a fuzzy ideal and its properties are investigated. It is shown that Zadeh s extension preserves strongly prime radicals. Also, a version of Theorem of Correspondence for strongly prime fuzzy ideals is proved. Keywords: radical Fuzzy ideal; Strongly prime ideal; Strongly prime radical; Fuzzy 1. Introduction In his pioneering paper, Zadeh 2 introduced the notion of a fuzzy subset A of a set X as a function from X to [0, 1]. In 1971, Rosenfeld 3 introduced fuzzy sets in the realm of group theory and formulated the concept of a fuzzy subgroup of a group. Since then, many researchers have been engaged in extending the concepts/results of abstract algebra to a broader framework of the fuzzy setting. In 1973, Formanek 4 showed that if D is a domain and G is a suitable free product of group, then the group ring DG is primitive. In the same year, Lawrence in his master s thesis showed that a generalization of Formanek s result was possible, in which the domain could be replaced by a prime ring with a finiteness condition called strongly prime. Although, the condition of strongly prime was used for primitive group rings it became more interesting. As a consequence, in 1975, Lawrence and Handelman 5 began to This research was partially supported by the Brazilian Research Council (CNPq) under the process /
2 2 study the properties of strongly prime rings and then many results were discovered, for example, every prime ring may be embedded in a strongly prime ring; all strongly prime rings are nonsingular and only the Artinian strongly prime rings have a minimal right ideal. In 1991, Information Sciences published a paper of Malik and Moderson 6,7 about radicals in the fuzzy setting. They provided a definition for radical of fuzzy ideal and showed a characterization of the Jacobson radical in terms of fuzzy maximal ideals. Thus, the primary ideal could be defined based on radical and many results was proven in the fuzzy environment e.g., if I is a fuzzy primary ideal on a ring R, then I is a fuzzy prime ideal of R. This work provides the concept of Strongly Prime Radical of a fuzzy ideal based on the ideas developed in. 1 The properties of this radical and its relation with strongly prime fuzzy ideals are investigated. Also, it is proved a fuzzy version of Correspondence Theorem for Strongly Prime fuzzy ideals. 2. Preliminaries This section explains some definitions and results that will be required in the next section. All rings are associative, with identity and usually denoted R. Definition 1. 8 A prime ideal in an arbitrary ring R is any proper ideal P of R such that, whenever I, J are ideals of R with a IJ P, either I P or J P. Equivalently P is prime whenever b xry P for some x, y R, then x P or y P. A prime ring is a ring in which (0) is a prime ideal. Note that a prime ring must be nonzero. Also, in commutative rings, the definition above is equivalent to: if ab P, then a P or b P. Thus, if P has the latter property, then it is called completely prime. In arbitrary rings, every completely prime is prime, but the converse is not true according. 1 In some papers, completely prime means strongly prime, but these concepts are different if you apply them to noncommutative rings. Every completely prime is strongly prime, but the converse is not true. The next definitions are necessary to build up the concept of strongly prime ideal. Definition 2. 5 Let A be a subset of a ring R. The right annihilator of A is written as follows An r (A) = {r R : Ar = (0)}. a IJ = {x : x = ij, i I, j J} b xry = {xry : r R}
3 3 Definition 3. 9 A ring R is said to be right strongly prime if each nonzero ideal I of R contains a finite subset F which has right annihilator zero. F is called right insulator. In other words, a right insulator in a ring R is a finite subset F R such that F r = 0, r R, implies r = 0. The ring R is said to be right strongly prime if every nonzero ideal(two-sided) contains an insulator. The left strongly prime is defined analogously. Handelman and Lawrence 5 gave an example to show that these two concepts are distinct. From this point forward, we call right strongly prime shortly strongly prime(sp). Definition 4. The ideal I of a ring R is strongly prime iff for every x R I there exists a finite subset F of R such that if r R and xf r I, then r I. It is not hard to prove that an ideal I of a ring R is strongly prime iff R/I is strongly ring. Let f : R S be a homomorphism of rings. We set sp(r) the set of all strongly prime ideals of R and sp f (R) = {I sp(r) : I Ker(f)}. The following theorem shows that there is a one to one correspondence between sp f (R) and sp(s). Theorem Let f : R S be an epimorphism of rings. Then (i) f(i) sp(s) for any I sp f (R); (ii) f 1 (I) sp f (R) for any I sp(s); (iii) The mapping Ψ : sp f (R) sp(s), Ψ(I) = f(i) is bijective. Definition 5 (Zadeh s Extension). Let f be a function from set X into Y, and let µ be a fuzzy subset of X. Define the fuzzy subset f(µ) by y Y {µ(x) : x X, f(x) = y}, if f 1 (y) f(µ)(y) = 0, otherwise. If λ is a fuzzy subset of Y, we define the fuzzy subset of X by f 1 (λ) where f 1 (λ)(x) = λ(f(x)). Let µ be any fuzzy subset of a set S and let α [0, 1]. The set {x X : µ(x) α} is called a level subset of µ which is symbolized by µ α. Clearly, if t > s, then µ t µ s. A fuzzy subset I of a ring R is called a fuzzy ideal of R if for all x, y R the following requirements are met: 1) I(x y) I(x) I(y); 2) I(xy) I(x) I(y). Note that 11 a fuzzy subset I of a ring R is a fuzzy ideal of R iff the level subsets I α, (α [0, 1]), are ideals of R.
4 4 3. Strongly radical of a fuzzy ideal The right strongly prime radical of a ring R is defined to be the intersection of all right strongly prime ideals of R. The dual notion of left strongly primeness determines a left strongly prime radical. An example given by Parmenter, Passman and Stewart 12 showed that these two radicals are distinct. In this section we define the concept of right strongly radical(shortly strongly radical) of a fuzzy ideal. Also, it is shown a fuzzy version of Correspondence Theorem and the right strongly prime radical(shortly SP radical) of a fuzzy ideal is defined and investigated. Throughout this section, unless stated otherwise, R has identity. Proposition 1. Let f : R S be a epimorphism of rings such that f 1 (Y ) is a finite set for all Y S. If I is a fuzzy set of R and J a fuzzy set of S, then f(i α ) = f(i) α and f 1 (J α ) = f 1 (J) α. Proof. Consider f(i α ) = {y S : y = f(x), x I α } and f(i) α = {y S : f(i)(y) α}. Let y f(i α ), y = f(x 0 ) where I(x 0 ) α. Thus, f(i)(y) = sup{i(x) : f(x) = y} I(x 0 ) α and then y f(i) α. On the other side, let y f(i) α, i.e. f(i)(y) = sup{i(x) : f(x) = y} α. As f is surjective, there exists x 0 R, where α I(x 0 ) sup{i(x) : f(x) = y} = f(i)(y). Thus, x 0 I α and then f(x 0 ) = y f(i α ). To prove f 1 (J α ) = f 1 (J) α, let x f 1 (J α ), then f(x) J α. Thus, f 1 (J)(x) = J(f(x)) α and, therefore, x f 1 (J) α. Now let x f 1 (J) α then J(f(x)) = f 1 (J)(x) α and therefore f(x) J α. In this case, it is not necessarily used f 1 (Y ) as a finite set. Theorem 2. Let f : R S be a epimorphism of rings such that f 1 (Y ) is a finite set for all Y S. If I is a SP fuzzy ideal of R such that Ker(f) I α for I(1) < α I(0), then f(i) is SP fuzzy ideal of R. Proof. Let I be a SP fuzzy ideal of R, where I α sp f (R) for I(1) < α I(0). Applying Theorem 1, (i) f(i α ) sp(s). By the Proposition 1 f(i) α is SP fuzzy ideal of S. Thus, f(i) sp(s). Proposition 2. Let f : R S be an epimorphism of rings. If J is a SP fuzzy ideal of S, then f 1 (J) is a SP fuzzy ideal of R, where f 1 (J) α Ker(f) for J(1) < α J(0). Proof. It is a consequence from proposition 1 and theorem 1 (ii).
5 5 For the next result, consider SP f (R) = {I is SP fuzzy ideal of R : I α sp f, I(1) < α I(0)} and SP (S) is the set of all SP fuzzy ideal of S. Theorem 3. (Correspondence Theorem) Let f : R S be an epimorphism of rings such that f 1 (Y ) is a finite set for all Y S. Then, there exists a bijection between SP f (R) and SP (S). Proof. Define Ψ : SP f (R) SP (S), Ψ(I) = f(i). Let I, M SP f (R), where I M. Thus, there exists x R, where I(x) M(x), if α = I(x), then I α M α. According to proposition 1 and Theorem 1, f(i) α = f(i α ) f(m α ) = f(m) α. Therefore, Ψ is injective. On the other hand, let J SP (S). As J α is SP by Theorem 1, we have f 1 (J α ) sp f (R), by Proposition 1, f 1 (J α ) = f 1 (J) α. Thus, f 1 (J) α is SP and f 1 (J) SP f (R). Moreover, Ψ(f 1 (J)) = f(f 1 (J)) = J. Therefore, Ψ is surjective. Definition 6. Given a crisp ideal I of a ring R, the strongly radical(or Levitzki radical) of I is s I = {P : P I, P is strongly prime}. Definition 7. Let I be a fuzzy ideal of R, the strongly radical of I is s I = P, where S I is the family of all SP fuzzy ideals P of R such P S I that I P. Clearly s I is an ideal, and if I is a SP fuzzy ideal, then s I = I Proposition 3. If I, J are a fuzzy ideal of a ring R, then: (i) if I J, then s I s J; (ii) s s I = s I; (iii) I α ( s I) α ; (iv) If I is SP fuzzy, then s I α = ( s I) α ; (v) s I J s I s J. Proof. (i) s J = P P = s I. (ii) It is easy to see that s I P S J P S I s s I. On the other side, let s show SI S s I. In fact, let P S I, then P I using (i) P = s P s I. (iii),(iv) and (v) is straightforward. Proposition 4. Let f : R S be a homomorphism of rings and I a fuzzy ideal of R. Then: 1)f(I) f( s I) s f( s I); 2) I f 1 ( s f(i)).
6 6 Proof. 1) Straightforward. 2) As f(i) s f(i), then f 1 (f(i)) f 1 ( s f(i)). Thus, I f 1 (f(i)) f 1 ( s f(i)). Proposition 5. Let f : R S be a homomorphism of rings and I a SP fuzzy ideal of R. Then, f( s I) s f(i). Proof. As I is SP fuzzy s I = I, then s f(i) = f( s s I). Thus, f( s I) s f( s I) = s f(i). Proposition 6. Let f : R S be an epimorphism of rings and I a SP fuzzy ideal of R, such that Ker(f) I α for I(1) < α I(0). Then, f( s I) = s f(i). Proof. As I is SP fuzzy ideal, I = s I, f(i) = f( s I). Using the theorem 2 f(i) is SP fuzzy ideal and then f(i) = s f(i). Thus, f( s I) = s f(i) = s s f(i). References 1. F. Boone Bergamaschi and R. Santiago, Strongly prime fuzzy ideals over noncommutative rings, in Fuzzy Systems (FUZZ), 2013 IEEE International Conference on, L. Zadeh, Information and Control 8, 338 (1965). 3. A. Rosenfeld, Journal of Mathematical Analysis and Applications 35, 512 (1971). 4. E. Formanek, Journal of Algebra 26, 508 (1973). 5. D. Handelman and J. Lawrence, Transactions of the American Mathematical Society 211, 209 (1975). 6. D. Malik and J. N. Mordeson, Information Sciences 53, 237 (1991). 7. D. S. Malik and J. N. Mordeson, Information sciences 65, 239 (1992). 8. K. R. Goodearl and R. B. Warfield Jr, An introduction to noncommutative Noetherian rings (Cambridge University Press, 2004). 9. M. Parmenter, P. Stewart and R. Wiegandt, Quaestiones Mathematicae 7, 225 (1984). 10. I. Herstein, Abstract algebra (1996). 11. V. Dixit, R. Kumar and N. Ajmal, Fuzzy Sets and Systems 49, 205 (1992). 12. M. Parmenter, D. Passman and P. Stewart, Communications in Algebra 12, 1099 (1984).
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