Skew monoid rings with annihilator conditions

Abstract

One of the most active and important research areas in noncommutative algebra is the investigation of skew monoid rings. Given a ring [Formula: see text] and a monoid [Formula: see text], we study the structure of the set of zero divisors and nilpotent elements in skew monoid ring [Formula: see text]. In the process we introduce a nil analog of the [Formula: see text]-skew [Formula: see text]-McCoy ring defined by Alhevaz and Kiani in [McCoy property of skew Laurent polynomials and power series rings, J. Algebra Appl. 13(2) (2014), Article ID: 1350083, 23pp.] and introduce the concept of so-called [Formula: see text]-skew nil [Formula: see text]-McCoy ring, which is a common generalization of [Formula: see text]-skew [Formula: see text]-McCoy rings, nil-McCoy rings and McCoy rings relative to a monoid. It is done by considering the nil-McCoy condition on a skew monoid ring [Formula: see text] instead of the polynomial ring [Formula: see text]. We also obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-skew nil [Formula: see text]-McCoy. Among other results, we prove that each regular [Formula: see text]-skew [Formula: see text]-McCoy ring [Formula: see text] is abelian (i.e. idempotents are central), where [Formula: see text] is any monoid with an element of infinite order and [Formula: see text] is a compatible monoid homomorphism. This answers, in a much more general setting, a question posed in [A. R. Nasr-Isfahani, On semiprime right Goldie McCoy rings, Comm. Algebra 42(4) (2014) 1565–1570], in the positive. Furthermore, we provide various examples and classify how the nil [Formula: see text]-McCoy rings behaves under various ring extensions.