SSP rings and modules (Q2806106)

The authors define \(\mathcal{A}\)-SSP modules, a generalisation of SSP modules and compare them with already known \(\mathcal{A}\)-\(C_i\) modules for \(i=2, 3\). Furthermore, they characterize SSP modules (a module \(M\) is an SSP module if the sum of any two direct summands of \(M\) is also a direct summand of \(M\)) and SSP rings (a ring \(R\) is an SSP ring if \(R_R\) is an SSP module). For instance, it is shown that:NEWLINENEWLINE{\parindent=6mm \begin{itemize}\item[1)] if \(P\) is a quasi-projective retractable module and its Jacobson radical \(J(P)\) is an essential submodule of \(P\), then \(P\) is an SSP module if and only if \(\text{End}_R(P)\) is a normal ring; \item[2)] if \(R\) and \(S\) are both normal rings (or both right semiartinian rings) and \(K=\begin{pmatrix} R & M \cr N & S\cr \end{pmatrix}\) is a formal matrix ring, then \(K\) is an SSP ring if and only if \(R\) and \(S\) are both SSP rings and NEWLINE\[NEWLINE\text{Reg}(K)=\begin{pmatrix} \text{Reg}(R) & M\cr N & \text{Reg}(S)\cr \end{pmatrix}.NEWLINE\]NEWLINENEWLINENEWLINE\end{itemize}}