On annihilations of ideals in skew monoid rings (Q2813358)

Skew model rings \(R*M\) of a monoid \(M\) over an unital ring \(R\), subject to a monoid homomorphism \(\sigma:M\to \text{End}\), are considered. The authors' nonzero right annihilator is right bounded, i.e., it contains a nonzero two-sided ideal. This notion was introduced by \textit{S. U. Hwang} et al. [Glasg. Math. J. 51, No. 3, 539--559 (2009; Zbl 1198.16001)]. Certain necessary and certain sufficient conditions on \(R\) and \(M\) are determined for \(R*M\) to be a strongly right AB-ring.NEWLINENEWLINE All of the main results are proved under the hypothesis that \(M\) is a unique product monoid. For example, if \(R\) is a nil-reversible ring (meaning that, if \(a\in R\) and \(b\in\text{nil}(R)\), then \(ab=0\) if and only if \(ba=0\)) and \(R\) is \(M\)-compatible, then \(R*M\) is a strongly right AB-ring. Here, \(M\)-compatibility means that \(ab=0\) if and only if \(a(\sigma(m)(b))=0\), for every \(a,b\in R\) and \(m\in M\).NEWLINENEWLINE In another direction, it is shown that under some additional hypotheses, the strong right AB-property on \(R*M\) implies that \(R*M\) has the right property (A), introduced in the noncommutative setting by \textit{C. Y. Hong} et al. [J. Algebra 315, No. 2, 612--628 (2007; Zbl 1156.16001)].NEWLINENEWLINE The latter property is concerned with right annihilators of two-sided ideals and it is a generalization of the notion introduced by \textit{J. A. Huckaba} and \textit{J. M. Keller} [Pac. J. Math. 83, 375--379 (1979; Zbl 0388.13001)] for the class of commutative rings. Relations between the strong right AB-property and the \(M\)-Armendariz and \(M\)-McCoy properties are also studied for the considered class of skew monoid rings \(R*M\).