\(\alpha\)-skew \(\pi\)-McCoy rings (Q1789849)

Summary: As a generalization of \(\alpha\)-skew McCoy rings, we introduce the concept of \(\alpha\)-skew \(\pi\)-McCoy rings, and we study the relationships with another two new generalizations, \(\alpha\)-skew \(\pi_1\)-McCoy rings and \(\alpha\)-skew \(\pi_2\)-McCoy rings, observing the relations with \(\alpha\)-skew McCoy rings, \(\pi\)-McCoy rings, \(\alpha\)-skew Armendariz rings, \(\pi\)-regular rings, and other kinds of rings. Also, we investigate conditions such that \(\alpha\)-skew \(\pi_1\)-McCoy rings imply \(\alpha\)-skew \(\pi\)-McCoy rings and \(\alpha\)-skew \(\pi_2\)-McCoy rings. We show that in the case where \(R\) is a nonreduced ring, if \(R\) is 2-primal, then \(R\) is an \(\alpha\)-skew \(\pi\)-McCoy ring. And, let \(R\) be a weak \((\alpha,\delta)\)-compatible ring; if \(R\) is an \(\alpha\)-skew \(\pi_1\)-McCoy ring, then \(R\) is \(\alpha\)-skew \(\pi_2\)-McCoy.