On cogenerator rings as \(\Phi\)-trivial extensions (Q1094517)

Let R be a ring with identity, M an (R,R)-bimodule and \(\Phi\) : \(M\otimes_ RM\to R\) an (R,R) bilinear map. Using \(\Phi\) the author defines a ring \(\Lambda\) on the abelian group \(R\oplus M\). This ring is called a \(\Phi\)-trivial extension of R by M. The author proves some necessary and sufficient condition for \(\Lambda\) to be a right cogenerator ring under the assumption that Im \(\Phi\) is nilpotent. Let \(\Gamma =\left( \begin{matrix} S\\ U\end{matrix} \begin{matrix} 0\\ T\end{matrix} \right)\) be a generalized triangular matrix ring, where both S and T are rings with identity and U a (T,S)-bimodule. The author shows that \(\Gamma\) is a right injective cogenerator ring if and only if both S and T are right injective cogenerator rings and \(U=0\). In case of \(S=T\) in \(\Gamma\), there holds that \(\Gamma\) is a right cogenerator ring if and only if T is a right cogenerator ring and \(U=0\). But when \(S\neq T\) in \(\Gamma\), the problem is unsolved.