Unmixedness theorem

Statement

Let $R$ be a commutative unital ring. If $I=(x_{1},x_{2},\ldots ,x_{n})$ is an ideal generated by $n$ elements such that the codimension of $I$ is $n$, then all minimal primes of $I$ have codimension $n$. If $R$ is a Cohen-Macaulay ring, then every associated prime of $I$ is minimal over $I$.