Let $M$ be maximal ideal in $R[x]$, the ring of polynomials over a commutative ring $R$, and let $P=M∩R$. If $a_1,...,a_n$ is a maximal R-sequence in $P$, it is clearly an R-sequence in $PR[x]$. Let $I=(a_1,...,a_n)$ and consider the image of $M$ in the ring $R[x]/IR[x]≅(R/I)[x]$, namely $\bar M=M/IR[x]$ which is a maximal ideal in $(R/I)[x]$ and so, can not consist entirely of zero-divisors. Thus the grade of $\bar M≥1$. My question: "why the grade of $M≥n+1$?"
Thanks for any help!
$\overline M$ contains a regular element on $R[x]/(a_1,\dots,a_n)R[x]$, say $\overline f$. Then $a_1,\dots,a_n,f$ is an $R[x]$-sequence in $M$.