Endo-prime ideals of commutative rings

Abstract

In this paper, we introduce a new broad generalization of prime ideals in commutative rings using endomorphisms defined on them. Let $R$ be a commutative ring with $1\neq 0$, and let $f:R\rightarrow R$ be a ring endomorphism. An ideal $I$ of $R$ is called an $f$-prime ideal if, whenever $a,b\in R$ and $ab\in I$, then $a\in I$ or $f(b)\in I$. Firstly, we study various basic properties of $f$-prime ideals. We show that this type of ideals can be useful in detecting the prime property of ideals in commutative rings. In particular, we give some new necessary and sufficient conditions for an ideal to be a prime ideal in terms of $f$-prime ideals. We prove through examples and results that these ideals enjoy some exotic features that prime ideals lack. In particular, we fully characterize rings that admit $f$-prime ideals, rings in which only $(0)$ is an $f$-prime ideal, and rings in which every ideal is an $f$-prime ideal. Finally, we relate $f$-prime ideals of a ring $R$ with other such ideals of various algebraic constructions on $R$.