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 (Formula presented.) (i.e., an R-linear map (Formula presented.) from M to R), an ideal I of R is said to be a (Formula presented.) -prime ideal if whenever (Formula presented.) and (Formula presented.) such that (Formula presented.), then (Formula presented.) or (Formula presented.). 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 (Formula presented.) -prime for every (Formula presented.) if and only if I is a trace ideal, and if the module M is the dual of I, then I is (Formula presented.) -prime for every (Formula presented.) 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.