Weakly special classes of semiprime rings (Q5932659)

For a class \(\mathcal M\) of rings (associative, but not necessarily with 1), let \(\mathcal M_\infty\) denote the class of all finite subdirect products of rings in \(\mathcal M\). In a previous paper, the author studied uniformly strongly prime rings, i.e. rings \(R\) which possess a finite subset \(X\subset R\) such that \(aXb=0\) implies \(a=0\) or \(b=0\) for all \(a,b\in R\). An overring \(S\) of \(R\) is called a tight extension if every non-zero one-sided ideal of \(S\) meets \(R\) non-trivially. The main theorem states that for the class \(\mathcal M \) of uniformly strongly prime rings, the class \(\mathcal M_\infty\) is closed under tight extensions. Some other properties of the relationship between classes \(\mathcal M \) of semiprime rings and \(\mathcal M _\infty\) are investigated.