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 $$I$$.