ϕ-s-prime Ideals Of Commutative Rings

Abstract

The concept of prime ideals and its generalizations have a distinguished place in commutative algebra since they are not only used in the characterization of various types of rings, but they also have some applications in other areas such as Graph Theory, Cryptology, Topology, Algebraic Geometry, etc. This paper aims to introduce and study ϕ-S-prime ideals of commutative rings which is a new generalization of prime ideals. Let R be a commutative ring with unity, S be a multiplicatively closed subset of R and ϕ:L(R)→L(R)∪{∅} be a function, where L(R) is the lattice of all ideals of R. An ideal I of R is said to be a ϕ-S-prime ideal if there exists a uniform s∈S such that ab∈I-ϕ(I) for some a,b∈R imply that sa∈I or sb∈I. In fact, prime ideals and its many recent generalizations such as S-prime ideals, weakly S-prime ideals and almost S-prime ideals are particular cases of our new concept. In this study, among other things, we determine the relations between ϕ-S-prime ideals and other classical ones. Also, we investigate the behavior of ϕ-S-prime ideals under rings homomorphisms, in factor rings, in quotient rings, in cartesian product of rings, in trivial extension. Finally, as an application of ϕS-prime ideals, we use them to characterize some special rings.