Minimal prime ideal

Remarks

Example: in $R = k[x, y]/(xy)$ with $k$ a field, the ideal $(x)$ is a minimal prime. It is prime because $R/(x) \cong k[y]$ is a domain. A prime $Q \subseteq (x)$ contains $xy = 0$, so it contains $x$ or $y$, and $y \notin (x)$ forces $x \in Q$. Non-example: $2\Z$ is prime in $\Z$, but it is not minimal, since the prime ideal $(0)$ is strictly smaller. By Zorn's lemma, every prime ideal containing $I$ contains a minimal prime over $I$; the key point is that the intersection of a chain of prime ideals is prime. The minimal primes over $I$ are the minimal primes of $R/I$, pulled back to $R$.