Preface . . . . . . . . . . . . . . . . . . . . . . . iii 1. Rings and Ideals . . . . . . . . . . . . . . . . . . . 1 2. Prime Ideals . . . . . . . . . . . . . . . . . . . . 6 3. Radicals . . . . . . . . . . . . . . . . . . . . . . 10 4. Modules . . . . . . . . . . . . . . . . . . . . . . 14 5. Exact Sequences . . . . . . . . . . . . . . . . . . . 20 6. Direct Limits . . . . . . . . . . . . . ....

The rings considered in this article are commutative with identity $1neq 0$. By a proper ideal of a ring $R$,  we mean an ideal $I$ of $R$ such that $Ineq R$.  We say that a proper ideal $I$ of a ring $R$ is a  maximal non-prime ideal if $I$ is not a prime ideal of $R$ but any proper ideal $A$ of $R$ with $ Isubseteq A$ and $Ineq A$ is a prime ideal. That is, among all the proper ideals of $R$,...

We investigate the structure of prime ideals of finite height in polynomial extension rings of a commutative unitary ring R. We consider the question of finite generation of such prime ideals. The valuative dimension of prime ideals of R plays an important role in our considerations. If X is an infinite set of indeterminates over R, we prove that every prime ideal of R[X] of finite height is fi...

In the present paper, we introduce the notion of fuzzy Aideals of MV -modules over PMV -algebras and several properties of fuzzy A-ideals are given. Using this concept, a prime fuzzy A-ideal is defined. Using a level set of a fuzzy set in MV -modules, we give a characterization of prime fuzzy A-ideals in MV -modules. Finally, we prove that the homomorphic image and preimage of prime fuzzy A-ide...

After the introduction of fuzzy sets by Zadeh [20] in 1965, several researchers have been working on the generalization of the fuzzification. In 1986, Atanassov [2] introduced the notion of intuitionistic fuzzy sets. For more details on the intuitionistic fuzzy sets, we refer to [2-4]. Fuzzy sets are intuitionistic fuzzy sets but the converse is not necessarily true [18]. Kim and Jun [7, 8] int...

The purpose of this paper is to study some properties of cartesian product of fuzzy completely prime ideals and fuzzy completely semiprime ideals of Γ-semigroups. Operator semigroups of a Γ-semigroup has been made to work by obtaining various relationships between cartesian product of fuzzy ideals (fuzzy completely prime ideals, fuzzy completely semiprime ideals) of a Γ-semigroup and that of it...