On cogenerator rings as $\phi$-trivial extensions
On cogenerator rings as $\phi$-trivial extensions
Let $R$ be a ring with identity and $M$ an $(R, R)$ -bimodule with a pairing $\Phi=[-$ , -$]$ : $M\otimes_{R}M\rightarrow R$ , that is, an $(R, R)$ -bilinear map satisfying $[m, m^{\prime}]m^{\prime\prime}$ $=m[m^{\prime}, m]$ .Then by defining a multiplication on the abelian group $R\oplus M$ as $(r, m)(r^{\prime}, m^{\prime})=(rr^{\prime}+[m, …