Associated Prime Ideal and Minimal Prime Ideal of an Ideal of an L-Subring

Abstract

In this paper, a systematic theory for the ideals of an L-ring has been developed. Earlier the authors have introduced the concepts of prime ideals, semiprime ideals, primary ideals, and radical of an ideal in an Lring. They have also introduced and discussed the notion of a maximal ideal in different papers wherein several results pertaining to these notions have been proved. In this paper, the concepts of associated prime ideal, minimal prime ideal, and that of irreducibility of an ideal in an L-ring have been introduced, which in fact, are the continuation of authors' previous works in the theory of ideals in L-rings. KEYWORDSprime ideal; semiprime ideal; primary ideal; radical of an ideal; L-subring S o far in the studies of fuzzy algebraic substructures, the definition of a subalgebra has been formulated in the framework of classical algebra. In fact, from Rosenfeld [1] onwards, almost all the researchers defined and studied the notion of a fuzzy subgroup of an ordinary group. Similar is the situation in the theory of fuzzy rings or lattice valued fuzzy rings [2−6] . Here we mention that such studies were initiated by Liu [7] . In Refs. [8,9] and in the theory of L-subgroups [10−17] , there is a deviation from this approach. Wu [18] gave a hint of pursuing studies of a fuzzy subgroup in a fuzzy group by providing the definition of a fuzzy normal subgroup in a fuzzy group. This idea was further taken up by Martinez [19,20] . In fact, in Ref.[10], we studied the notion of a normal fuzzy subgroup of a fuzzy group and defined a characteristic fuzzy subgroup of a fuzzy group. This laid the foundation of the theory of fuzzy subalgebras where the parent structures are also fuzzy algebras. The development of fuzzy algebraic structures was hampered mainly because of two reasons. Firstly, it was due to some inherent problem that the analogs of some of the concepts and results of classical algebra were not even formulated in this setting such as transitivity of properties of subsets of an algebra. For example: "a normal subgroup of a characteristic subgroup of a group is a normal subgroup of the given group"; and "a prime ideal of an ideal of a ring is an ideal of the given ring". Moreover, the concept of maximality of a fuzzy substructure of an algebraic structure is not properly formulated. Secondly, since most of the concepts studied in fuzzy algebraic