Supernilpotent radicals of \(\Gamma_ N\)-rings (Q1365434)

Let \(\mathcal P\) be a weakly special class of rings. Then \(\mathcal P\) is said to satisfy condition (*) if \(R\in{\mathcal P}\) implies \(eRe\in{\mathcal P}\) for every idempotent element \(e\) of \(R\). A \(\Gamma\)-ring \(M\) with right operator ring \(R\) is called a \(\mathcal P\)-\(\Gamma\)-ring if \(R\in{\mathcal P}\) and \(M\Gamma x=0\) implies \(x=0\), for all \(x\in M\). The class \({\mathcal P}(\Gamma)\) of \(\mathcal P\)-\(\Gamma\)-rings is weakly special. The upper radical determined by this class is denoted \({\mathcal R}_{{\mathcal P}(\Gamma)}\). An ideal \(I\) of \(M\) is called a \(\mathcal P\)-ideal of \(M\) if \(M/I\) is a \(\mathcal P\)-\(\Gamma\)-ring. In this paper, it is shown that if \(\mathcal P\) is a weakly special class of rings which satisfies condition (*), the \(\mathcal P\)-ideals of the matrix \(\Gamma_{nm}\)-ring \(M_{mn}\) are precisely the sets \(I_{mn}\), where \(I\) is a \(\mathcal P\)-ideal of \(M\). Consequently, \({\mathcal R}_{{\mathcal P}(\Gamma)}(M_{mn})=({\mathcal R}_{{\mathcal P}(\Gamma)}(M))_{mn}\). The author then turns his attention to the Morita context ring \(\left[\begin{smallmatrix} R &\Gamma \\ M &L\end{smallmatrix}\right]\) where \(M\) is a \(\Gamma_N\)-ring, and similar natural results are obtained in this case.