# Nullspace of a matrix modulo an ideal

Suppose $R$ is a multivariate polynomial ring and $I$ is an ideal in $R$.

Let $M$ be a $n\times n$ square matrix with entries in $R$, and suppose that det($M$) lies in $I$. Thus, $M$ has a non-trivial nullspace when the variables take values in the variety defined by $I$.

Is there an algorithm for producing a non-trivial $n$-vector $v$, with entries in $R$, such that the entries of $M v$ belong in $I$?