A new class of non-semiprime quasi-Armendariz rings. (Q2875468)

Let \(R\) be an associative ring with identity and let \(R[x]\) denote the polynomial ring over \(R\). The ring \(R\) is called a quasi-Armendariz ring if whenever \(f(x),g(x)\in R[x]\) satisfy \(f(x)R[x]g(x)=0\), we have \(aRb=0\) for every coefficient \(a\) of \(f(x)\) and every coefficient \(b\) of \(g(x)\).NEWLINENEWLINE \textit{Y. Hirano} [J. Pure Appl. Algebra 168, No. 1, 45-52 (2002; Zbl 1007.16020)] proved that the semiprime rings are quasi-Armendariz. In this paper the author studies a large class of non-semiprime quasi-Armendariz rings. Following \textit{Z. Liu} and \textit{R. Zhao} [Glasg. Math. J. 48, No. 2, 217-229 (2006; Zbl 1110.16003)], an ideal \(I\) of \(R\) is said to be left \(s\)-unital if for each \(a\in I\) there is an \(x\in I\) such that \(xa=a\). A ring \(R\) is called right APP-ring if for every element \(a\in R\), the right annihilator of \(aR\) is left \(s\)-unital as an ideal of \(R\).NEWLINENEWLINE Let \(M\) be the monoid generated by \(u\) with the relation \(u^n=0\), where \(n\geq 2\) is a natural number. Denote by \(R[M]\) the monoid ring of \(M\) over \(R\). The main result of the paper asserts that if \(R\) is a right APP-ring, then \(R[M]\) is a non-semiprime quasi-Armendariz ring. Thus, if \(R\) is a right APP-ring, then \(R[x]/(x^n)\) is a quasi-Armendariz rings with non-zero nilpotent ideal.