On Jacobson property of \(\Gamma_ N\)-rings (Q1362812)

Let \(\mathcal P\) be a class of prime rings such that \(R\in{\mathcal P}\) if and only if \(eRe\in{\mathcal P}\) for any nonzero idempotent element \(e\) of \(R\). If \(M\) is \(\Gamma\)-ring with right operator ring \(R\), then \(M\) 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 of all \({\mathcal P}\)-\(\Gamma\)-rings is denoted \({\mathcal P}_{(\Gamma)}\). An ideal \(I\) of \(M\) is called a \(\mathcal P\)-ideal if \(M/I\in{\mathcal P}_{(\Gamma)}\). The \(\mathcal P\)-\(\Gamma\) radical of \(M\) is the intersection of the \(\mathcal P\)-ideals of \(M\). For the case that \(M\) is a \(\Gamma_N\)-ring with a right unity, a natural characterization is given for the \(\mathcal P\)-ideals of the ring \(M_2=\left[\begin{smallmatrix} R &\Gamma\\ M &L\end{smallmatrix}\right]\). This leads to a characterization of the \(\mathcal P\)-radical of \(M_2\). A \(\Gamma\)-ring \(M\) is called a \(\mathcal P\)-Jacobson \(\Gamma\)-ring if the prime and \(\mathcal P\)-radicals coincide in all proper homomorphic images of \(M\). It is shown that \(M\) is a \(\mathcal P\)-ring if and only if \(M_{mn}\) is a \(\mathcal P\)-\(\Gamma_{nm}\)-ring. This gives an obvious characterization of the \(\mathcal P\)-radical of \(M_{mn}\). Moreover \(M\) is a \(\mathcal P\)-Jacobson \(\Gamma\)-ring if and only if \(M_{mn}\) is a \(\mathcal P\)-Jacobson \(\Gamma_{nm}\)-ring. The final section of the paper gives some results concerning the right operator ring \(R\) of a \(\Gamma\)-ring \(M\). \(M\) is a \(\mathcal P\)-Jacobson \(\Gamma\)-ring if and only if \(R\) is a \(\mathcal P\)-Jacobson ring. Let \(\alpha\) and \(\gamma\) denote the radical classes of \(\mathcal P\)-Jacobson rings and \(\mathcal P\)-Jacobson \(\Gamma\)-rings, respectively. If \(M\) has a left unity, then \(\alpha(R)\subseteq(\gamma(M))^*\), and if \(M\) also has a right unity, then \(\gamma(M)\subseteq(\alpha(R))^*\). If \(M\) is a \(\Gamma_N\)-ring with a right unity, then \(M_2\) is a \(\mathcal P\)-Jacobson ring if and only if \(M\) is a \(\mathcal P\)-Jacobson \(\Gamma\)-ring. A natural characterization is provided for \(\alpha(M_2)\) in the case that \(M\) has left and right unities.