PS and CESS Property of Formal Triangular Matrix Rings

Abstract:

      Let $R$ be a ring. Recall that a right $R$-module $M$ ($R_R$, resp.) is said to be a PS-module (PS-ring, resp.) if it has projective socle. $M$ is called a CESS-module if every complement summand in $M$ with essential socle is a direct summand of $M$. We show that the formal triangular matrix ring $T=\left( \begin{array}{cc} A & 0 \\ M& B \\ \end{array} \right)$ is a PS-ring if and only if $A$ is a PS-ring, $M_{A}$ and $l_{B}(M)=\{b\in B\mid bm=0, \forall m\in M\}$ are PS-modules and $\Soc(l_{B}(M))\bigotimes M=0$. Using the alternative of right $T$-module as triple $(X,Y)_{f}$ with $X\in {\rm Mod}$-$A$, $Y\in {\rm Mod}$-$B$ and $f:Y\bigotimes M\rightarrow X$ in ${\rm Mod}$-$A$, we show that if $T_T$ is a CESS-module, then $A_A$ and $M_A$ are CESS-modules.