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 $\varphi$ (i.e., an R-linear map $\varphi$ from M to R), an ideal I of R is said to be a $\varphi$-prime ideal if whenever $a\in R$ and $m\in M$ such that $a\varphi (m)\in I$, then $a\in I$ or $\varphi (m)\in I$. 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 $\varphi$-prime for every $\varphi \in {\mathit{\text{Hom}}}_R(I,R)$ if and only if I is a trace ideal, and if the module M is the dual of I, then I is $\varphi$-prime for every $\varphi \in {\mathit{\text{Hom}}}_R(I^{-1},R)$ 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.

Keywords:

• $\varphi$-prime ideal
• prime submodule
• Prüfer domain
• strongly divisorial ideal
• trace ideal
• valuation domain

2020 Mathematics Subject Classification:

• Primary: 13G05, 13A15
• Secondary: 13F05, 13F30