On graded 1-absorbing prime ideals
Let \(R\) be a commutative ring graded by a group. A graded 1-absorbing prime ideal of \(R\) is defined by the authors as being a proper graded ideal \(P\) such that for any non-invertible homogeneous elements \(x,y,z\in P\), either \(xy\in P\) or \(z\in P\). Several basic results and equivalent characterizations of such ideals are presented. It is showed that the graded prime ideal of a graded 1-absorbing prime ideal is graded prime. It is proved that if every nonzero proper graded ideal of \(R\) is graded 1-absorbing prime, then every homogeneous element \(x\) of \(R\) is \(\pi\)-regular, i.e., there exist a positive integer \(n\) and \(y\in R\) such that \(x^{2n}y=x^n\). A graded version of the prime avoidance theorem is presented.
scientific article; zbMATH DE number 7375699
scientific article