On $$\phi $$-1-absorbing prime ideals

Abstract

In this paper, we introduce $$\phi $$ -1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity $$1\ne 0$$ and $$\phi :\mathcal {I}(R)\rightarrow \mathcal {I}(R)\cup \{\emptyset \}$$ be a function where $$\mathcal {I}(R)$$ is the set of all ideals of R. A proper ideal I of R is called a $$\phi $$ -1-absorbing prime ideal if for each nonunits $$x,y,z\in R$$ with $$xyz\in I-\phi (I)$$ , then either $$xy\in I$$ or $$z\in I$$ . In addition to give many properties and characterizations of $$\phi $$ -1-absorbing prime ideals, we also determine rings in which every proper ideal is $$\phi $$ -1-absorbing prime.