# Questions tagged [local-rings]

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

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### Commutative ring satisfying a.c.c. and d.c.c. on radical ideals

Let $R$ be a commutative ring with unity whose prime spectrum is both Noetherian and Artinian under Zariski topology i.e. $R$ satisfies a.c.c. and d.c.c. on radical ideals. Then is it true that $R$ ...
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### Exercise 4.4.1 in Weibel's 'An Introduction to Homological Algebra'.

I can solve this question on the assumption that the $x_i$s are not zero-divisors since $\dim(R/(x)) = \dim(R)-1$ if $x$ is not a zero-divisor. My question is, how do I prove that they are not zero ...
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### Existence of ideal in Cohen-Macaulay ring, going modulo which still gives Cohen-Macaulay [closed]

Let $R$ be a local Cohen-Macaulay ring of dimension $\le 2$. Does there necessarily exist an ideal $J$ of $R$ such that $\sqrt J$ is a minimal prime ideal of $R$ and $R/J$ is Cohen-Macaulay ?
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### Quotient ring local if Ring is local

I want to show: If $R$ is local and $I\neq R$ an ideal, then $R/I$ is also local. We already know: A Ring $R$ is local if and only if $R-R^{\times} = \{r\in R \, | r \notin R^\times \}$ is an ideal. ...
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### When is a map of local rings finite?

Let $A,B$ be Noetherian local rings, and let $A \to B$ be a ring homomorphism such that the induced map $\operatorname{Spec} B \to \operatorname{Spec} A$ is surjective and quasifinite (of finite type ...
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### Commutative local ring with $10$ ideals

Let $R$ be a commutative ring with unity with exactly $10$ ideals (including $\{0\}$ and $R$ ) . Then is it true that $R$ is a Principal Ideal Ring ? My Work: I know that any commutative ring with $5$...
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### Valuation ring whose unique maximal ideal and every ideal of finite height is principal

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
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Consider two analytic algebras $\mathcal{O}_{\mathbb{C}^{m},0}/\mathcal{I}_{A}$ and $\mathcal{O}_{\mathbb{C}^{n},0}/\mathcal{I}_{B}$, where $\mathcal{I}_{A}:=\mathcal{O}_{\mathbb{C}^{m},0}f^{1}_{0}+\... 0answers 50 views ### Prime ideal of a polynomial ring restricted to the ring of coefficients This is a question following on from here. There I was asking about the proof of Proposition 6.6 in Chapter II of Hartshorne's text. At this point I'm still completely stuck. I think I have reduced ... 0answers 23 views ### Are the quotients in$I$-adic completions of a local Noetherian ring isomorphic for proper ideals$I$Let$R$be a local Noetherian ring with maximal ideal$m$. Now let$I$be a proper ideal of$R$and$\hat{R}_I$the$I$-adic completion of$R$. I have to prove that they have the same Samuel function. ... 0answers 47 views ### More aesthetically pleasing proof of lifting. Say you have a complete Noetherian local ring$(A,\mathfrak m_A,k_A)$and a complete Noetherian local$A$-algebra$B$with residue field$A$. Furthermore, suppose you can find a surjective local ... 1answer 89 views ### Local, factorization domain with principal maximal ideal is a PID? Let$R$be a Factorization domain (https://en.m.wikipedia.org/wiki/Atomic_domain ) which is local and with principal maximal ideal. Then is$R$a Bezout domain i.e. every finitely generated ideal is ... 0answers 24 views ### Difference between two localizations What is the difference between$\mathbb{C}[x]_{(x)}[y]$and$\mathbb{C}[x,y]_{(x)}$? To me they are both equal to:$\{ \frac{f(x, y)}{g(x,y)} | g(0, y) \neq 0 \}$Is this true and if it isn't can ... 1answer 40 views ### Statement of Lech's lemma The statement of the theorem$14.12$(page 110) in Commutative ring theory by Matsumura is: Let$A$be a$d$-dimensional local ring, and$x_1,\ldots ,x_d$be a system of parameters; set$\mathfrak q=(...
I am interested in a ring $R$ with an ideal $I\subset R$ and the following properties: 1) $R$ is complete with respect to $I$-adic topology; 2) $\cap_{n\geq 1} I^n=\{0\}$; 3) $R/I$ is finite. In ...
Let $R\subseteq S$ be two finite commutative local rings with unique maximal ideals $m$ and $M$, respectively. We say that $S$ is a separable extension of $R$ if $mS=M$. We also say that $S$ is a ...