Let $R$ be a Cohen-Macaulay ring and $M$ be a finite generated maximal Cohen-Macaulay module. I know that the R-regular sequence must be $M$-regular. Here are my questions:
1) Must an $M$-regular sequence also be $R$-regular?
2) If $M$ is not maximal, is an $R$-regular sequence also $M$-regular?
1) Consider $R=K[X,Y]/(X^2Y)$ and $M=R/(x)$. Then $M$ is MCM, $y$ is $M$-regular and $y$ is not $R$-regular.
2) Consider $R=K[X,Y]$ and $M=R/I$, where $I=(X)$. Then $X,Y$ is not $M$-regular.