Reducing system of parameters and the Cohen--Macaulay property

Let $R$ be a local ring and let ($x_1\biss x_r$) be part of a system of parameters of a finitely generated $R$-module $M,$ where $r < \dim_R M$. We will show that if ($y_1\biss y_r$) is part of a reducing system of parameters of $M$ with $(y_1\biss y_r)M=(x_1\biss x_r)M$ then $(x_1\biss x_r)$ is already reducing. Moreover, there is such a part of a reducing system of parameters of $M$ iff for all primes $P\in \supp M \cap V_R(x_1\biss x_r)$ with $\dim_R R/P = \dim_R M -r$ the localization $M_P$ of $M$ at $P$ is an $r$-dimensional \cm\ module over $R_P$. Furthermore, we will show that $M$ is a \cm module iff $y_d$ is a non zero divisor on $M/(y_1\biss y_{d-1})M$, where $(y_1\biss y_d)$ is a reducing system of parameters of $M$ ($d := \dim_R M$).