On weakly 1-absorbing prime ideals

Abstract

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $$A $$ be a commutative ring with a nonzero identity $$1\ne 0.$$ proper ideal $$P $$ of $$A $$ is said to be a weakly 1-absorbing prime ideal if for every nonunits $$x,y,z\in A $$ with $$0\ne xyz\in P, $$ then $$xy\in P$$ or $$z\in P. $$ In addition to give many properties and characterizations of weakly 1-absorbing prime ideals, we also determine rings in which every proper ideal is weakly 1-absorbing prime. Furthermore, we investigate weakly 1-absorbing prime ideals in C(X), which is the ring of continuous functions of a topological space $$X. $$