On pseudo Cohen-Macaulay modules

Let $(A,\frak m)$ be a commutative Noetherian local ring with the maximal ideal $\frak m$ and $M$ be a finitely generated $A$- module with $\dim\,M=d$. For each system of parameters ${\underline x} {}=(x_1,...,x_d)$ of $M,$ we are interested the submodule $Q_M ({\underlinex})=\bigcup_{n>0}\Big((x_1^{n+1},...,x_d^{n+1})M:x_1^n\cdots x_d^n\Big)If there exists a system of parameters ${\underline x}$ such that $e({\underline x};M)=\ell_A(M\big/Q_M({\underline x}))$ then $M$ is called {\smsl pseudo Cohen-Macaulay} (pCM for short). This paper presents some properties of pseudo Cohen-Macaulay modules in terms of local cohomology and system of parameters