On functional prime ideals in commutative rings

Abstract

In this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a ∈ R and m ∈ M such that a ϕ ( m ) ∈ I , then a ∈ I or ϕ ( m ) ∈ I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ ∈ Hom R ( I , R ) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ ∈ Hom R ( I − 1 , R ) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided.