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Fuzzy Primeness in Quantales
Abstract
This paper is an investigation about primeness in quantales environment. It is proposed a new definition for prime ideal in noncommutative setting. As a consequence, fuzzy primeness can be defined in similar way to ring theory.
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This paper is a step forward in the field of fuzzy algebra. Its main target is the investigation of some properties about uniformly strongly prime fuzzy ideals (USPf) based on a definition without a-cuts dependence. This approach is relevant because it is possible to find pure fuzzy results and to see clearly how the fuzzy algebra is different from classical algebra. For example: in classical ring theory an ideal is uniformly strongly prime (USP) if and only if its quotient is a USP ring, but as we shall demonstrate here, this statement does not happen in the fuzzy algebra. Also, we investigate the Zadeh's extension on USPf ideals.
In this paper it is defined the concept of strongly prime fuzzy ideal for noncommutative rings. Also, it is proved that the Zadeh's extension preserves strongly fuzzy primeness and that every strongly prime fuzzy ideal is a prime fuzzy ideal as well as every fuzzy maximal is a strongly prime fuzzy ideal.
A ring R is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple. All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring RG is SP, then R is SP and G contains no nontrivial locally finite normal subgroups. Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and R are the only torsion ideals.( 1 ^{1} ) A list of questions is appended.
Rough set theory is a useful mathematical tool for dealing with the uncertainty and granularity in information systems. In this paper, rough approximations are introduced into quantales, a kind of partially ordered algebraic structure with an associative binary multiplication. The notions of (upper, lower) rough (prime, semi-prime, primary) ideals of quantales are proposed and verified to be the extended notions of usual (prime, semi-prime, primary) ideals of quantales respectively. Global order properties of all lower (upper) rough ideals of a quantale are also investigated. The rough radical of an upper rough ideal in a quantale is defined and the prime radical theorem of rings is generalized to upper rough ideals of quantales. Some results about homomorphic images of rough ideals and rough prime ideals of semigroups and rings are also generalized and improved in the field of quantales.
Rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems. Motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups, rings and lattices, in this paper we introduce first the notions of fuzzy (prime, semi-prime, primary) ideals of quantales and investigate their properties. Several characterizations of such ideals are presented. Then, we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales, and establish relationships between these ideals and fuzzy prime ideals of quantales. By applying rough set theory to fuzzy ideals of quantales, we furthermore define rough fuzzy (prime, semi-prime, primary) ideals of quantales, generalizing Yang and Xu’s work on quantales to the fuzzy environment. Finally, relationships between the upper (resp. lower) rough fuzzy ideals of quantales and the upper (resp. lower) approximations of their homomorphic images are also discussed.
A characterization of rings which are both Artinian and Noetherian is obtained. They are proved to be the rings whose fuzzy ideals assume only finitely many distinct values. Prime fuzzy ideals have been defined and conditions have been obtained under which they have only two distinct values.
The class of all uniformly strongly prime rings is shown to be a special class of rings which generates a radical class which properly contains both the right and left strongly prime radicals and which is independent of the Jacobson and Brown-McCoy radicals.
The purpose of this paper is to extend the concept of a fuzzy set by introducing the concept of a fuzzy nil radical of a fuzzy ideal in a ring. Using this concept, a fuzzy primary ideal is defined. Some fundamental results concerning these notions are proved.
The concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings. This class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals. It is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary. A study of (i) the level ideals of a fuzzy semiprimary ideal (ii) the characteristic function of a semiprimary ideal (iii) the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and (iv) the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms is carried out.
The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals, and to prove some fundamental properties. In particular, it will give a characteristic of a field by a fuzzy ideal.
In this paper the concepts of a primary L-fuzzy ideal and a prime L-fuzzy ideal and P-primary are introduced and some fundamental propositions proved. In particular, a structural theorem for a primary L-fuzzy ideal belonging to a prime L-fuzzy ideal is given.
The merits and demerits of various definitions of prime and primary fuzzy ideals are discussed and new definitions are suggested. It is shown that the new definitions are more satisfactory. The nil radical is defined as against the (prime) radical of a fuzzy ideal. Some theorems regarding the nil radical are proved. It is shown that the two radicals coincide when the grade membership lattice is totally ordered.
The aim of this paper is to study fuzzy ideals of a ring, to characterise regular rings and Noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring Z of integers.
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
In this paper we give suitable definitions of the operations of L-fuzzy ideals in a ring, as intersection, addition, multiplication and residual division. And we prove some fundamental properties of the operations analogous to those that hold for the operations of ordinary ideals (see [3]).
The notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers. A procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established. All fuzzy prime (maximal) ideals of a given ring are determined by establishing a one-to-one correspondence between fuzzy prime (maximal) ideals and the pairs (P, α), where P is a prime (maximal) ideal of the ring and α is a prime element (dual atom) in the complete lattice.
In this paper we introduce prime fuzzy ideals over a noncommutative ring.
This notion of primeness is equivalent to level cuts being crisp prime ideals.
It also generalizes the one provided by Kumbhojkar and Bapat in [Not-so-fuzzy
fuzzy ideals, Fuzzy Sets and Systems 37 (1990), 237--243], which lacks this
equivalence in a noncommutative setting. Semiprime fuzzy ideals over a
noncommutative ring are also defined and characterized as intersection of
primes. This allows us to introduce the fuzzy prime radical and contribute to
establish the basis of a Fuzzy Noncommutative Ring Theory.