On functional prime ideals in commutative rings

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 aϕ(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∈HomR(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∈HomR(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.