Corollary 1

Tag 046B, permanent. Cite it as [Bourbaki, Tag 046B].

Commutative Algebra, chapter II, § 2. Rings and modules of fractions

Let $A$ be a ring and $S$ a multiplicative subset of $A$ . Every ideal $p$ of $A$ which is maximal among those which do not meet $S$ is prime.

By Proposition 11, the hypothesis on $p$ means that $p = (i_A^S)^{-1}(m')$ , where $m'$ is a maximal ideal of $S^{-1}A$ ; as $m'$ is prime, so is $p$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 11