What is a minimal prime ideal of a ring

Question

From Wikipedia:

A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note that we do not exclude I even if it is a prime ideal.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal.

$\{0\}$ is always a prime ideal, so wouldn't this mean that in every ring only $\{0\}$ is minimal? Because for any prime ideal $\mathfrak{p}$ one always has $\{0\}\subset\mathfrak{p}$.

Here is a further answer to your title question. – Dietrich Burde Jul 07 '16 at 15:21 — Dietrich Burde, Jul 07 '16 at 15:21

score 3 · Accepted Answer · answered Jul 07 '16 at 15:20

3

For a commutative ring $R$ it is not necessarily true that the zero ideal $0 \subseteq R$ is a prime ideal. This holds if and only if $R$ is an integral domain.

answered Jul 07 '16 at 15:20

Jendrik Stelzner

16,401

You are right, thank you for reminding me. – azureai Jul 07 '16 at 15:20

What is a minimal prime ideal of a ring

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