The Stacks project

Proof. We will use that $\bigcap \mathfrak m^ n = 0$ by Lemma 10.51.4. Let $f, g \in R$ such that $fg = 0$. Suppose that $f \in \mathfrak m^ a$ and $g \in \mathfrak m^ b$, with $a, b$ maximal. Since $fg = 0 \in \mathfrak m^{a + b + 1}$ we see from the result of Lemma 10.106.1 that either $f \in \mathfrak m^{a + 1}$ or $g \in \mathfrak m^{b + 1}$. Contradiction. $\square$

Comments (2)

Comment #8848 by Et on

Proposed simpler proof: by lemma 00IP, hence the map is an inclusion. By lemma 00NO is an integral domain, and hence so is .

Comment #9238 by on

This doesn't work because there is no ring map . A posteriori we see that there is a map of multiplicative monoids -- but this isn't true in general for Noetherian local rings.

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  • 6 comment(s) on Section 10.106: Regular local rings

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