List questions from localization

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$R=\{a/b: a,b \in \mathbb{Z}$ and $b$ is odd}. Show that the ring $R$ is a PID.

Published on 21 Oct 2015 - 0:20

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Localization of a one-dimensional noetherian integral domain

Published on 27 Mar 2026 - 21:21

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Prove that $a\mathbb{Z} \cap \mathbb{Z} \backslash p\mathbb{Z} \neq \varnothing \iff a\mathbb{Z}_{(p)}=\mathbb{Z}_{(p)}$

Published on 28 Oct 2015 - 0:29

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Prove that there exists a unique homomorphism of $\mathbb{Z}_{(p)}$-modules $\beta:M_p \rightarrow N$ such that $\beta(m/r)=\frac{1}{r}\alpha(m)$.

Published on 28 Oct 2015 - 22:09

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Show that each element of $K$ can be written as $\frac{b}{a}$ where, $b\in B, a\in A$

Published on 30 Oct 2015 - 12:25

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When is localization a quotient?

Published on 06 Nov 2015 - 10:18

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Is every local quasi-finite morphism finite?

Published on 07 Nov 2015 - 1:17

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A Notation in Matsumura's Commutative Algebra

Published on 07 Nov 2015 - 22:01

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For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$

Published on 07 Nov 2015 - 22:15

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groupoids and localization in categories

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Do isomorphic quotient fields imply isomorphic rings?

Published on 14 Nov 2015 - 12:45

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Localization $(R_{\mathfrak p})_{\mathfrak q}$ for an Artinian ring.

Published on 15 Nov 2015 - 23:55

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Localizations of an Artinian ring are isomorphic to quotients.

Published on 16 Nov 2015 - 1:16

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When is the localization of a commutative ring a finitely generated projective module?

Published on 29 Mar 2026 - 11:43

List Question

Serre quotient category

$R=\{a/b: a,b \in \mathbb{Z}$ and $b$ is odd}. Show that the ring $R$ is a PID.

Localization of a one-dimensional noetherian integral domain

Prove that $a\mathbb{Z} \cap \mathbb{Z} \backslash p\mathbb{Z} \neq \varnothing \iff a\mathbb{Z}_{(p)}=\mathbb{Z}_{(p)}$

Prove that there exists a unique homomorphism of $\mathbb{Z}_{(p)}$-modules $\beta:M_p \rightarrow N$ such that $\beta(m/r)=\frac{1}{r}\alpha(m)$.

Show that each element of $K$ can be written as $\frac{b}{a}$ where, $b\in B, a\in A$

When is localization a quotient?

Is every local quasi-finite morphism finite?

A Notation in Matsumura's Commutative Algebra

For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$

groupoids and localization in categories

Do isomorphic quotient fields imply isomorphic rings?

Localization $(R_{\mathfrak p})_{\mathfrak q}$ for an Artinian ring.

Localizations of an Artinian ring are isomorphic to quotients.

When is the localization of a commutative ring a finitely generated projective module?

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