形式三角矩阵环的PS性质和CESS性质

设$R$是环. 称右$R$-模$M$是PS-模,如果$M$具有投射的socle. 称$R$是PS-环,如果$R_R$是PS-模. 称$M$是CESS-模,如果$M$的任意具有基本socle的子模是$M$的某个直和因子的基本子模.本文给出了形式三角矩阵环 $T=\left( \begin{array}{cc} A & 0 \\

PS and CESS Property of Formal Triangular Matrix Rings

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.