Primeness relative to a homomorphism

Let R be a commutative ring and let X,M be two unitary R-modules. In this note we introduce the notion of a prime submodule relative to a homomorphism ϕ(ϕ-prime submodules). Several properties of ϕ-prime submodules are presented and we find two conditions under which ϕ-prime gives prime. Then its behavior with fractions formation on a multiplicatively closed subset S of R will be studied. When P is an invertible cyclic submodule of M, we characterize the primness of P in terms of ϕ-primness of P for all ϕ∈HomR(P−1,M), and for all ϕ∈HomR(EndR(P),M). Then, we consider maximal ϕ-prime (ϕ-primary) submodules and see that maximal ϕ-primary and maximal ϕ-prime coincide. The results of this paper are initiated from the paper [A. Mimouni, On functional prime ideals in commutative rings, Comm. Alg. 52(11), 2024, 4525–4533]. Communicated by Scott Chapman