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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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Why is $F[x]/(x^n)$ a local ring?

How is $\frac{F[x]}{(x^n)}$ a local ring? I was trying to show the elements which are not units are nilpotent. But not being able to prove it properly. Please give some hint.
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How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?

Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
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A sufficient condition for a unitary ring to be a local ring

Theorem. Let $R$ be a unitary ring such that $R$ is a subring of a division ring $D$. If for all $d(\ne 0)\in D$ either $d\in R$ or $d^{-1}\in R$ then $R$ is a local ring. My Proof. It suffices ...
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Proving that $\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a complete intersection ring

I have found in some sources that the ring $R=\mathbb{F}_p[x_1,\dots,x_n]/(x_1^p,\dots,x_n^p)$ is a local complete intersection ring. I need this result in order to apply a related theorem and I ...
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Betti numbers of finitely generated module over Noetherian local ring, after going modulo a regular element

For a finitely generated module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ , let $b_i^R(M):= \dim_k \operatorname{Tor}_i^R (k,M)$. It is known that this $i$-th Betti numbers ...
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In a commutative local Noetherian ring $R$ with maximal ideal $J$, if $J$ is not nilpotent then $R$ is an integral domain.

We've just proved this result: Let $R$ be a commutative, local, Noetherian ring. Suppose that $J$ (the maximal ideal) is principal. Then every nonzero ideal of $R$ is a power of $J$. And now we ...
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Dimension of $m/m^2$ over $k$ where $m=m_P(V)$ and $V=V(x^2-y^3,y^2-z^3)$

Consider the variety $V=V(x^2-y^3,y^2-z^3)$ and coordinate ring $\Gamma(V)=\frac{k[x,y,z]}{I(V)}$. Let $k(V)$ be the field of fractions of $\Gamma(V)$. Let $P=(0,0,0)$. An element $f\in k(V)$ with $f=\...
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What is meant by $\operatorname{Tor}(P,K)$ for $P$ a left $R$-module?

Let $R$ be a finite dimensional algebra over a field $K$. Suppose there is a two-sided ideal $I$ of $R$ such that (a) $R=K\oplus I$; (b) $I$ is nilpotent. Question If $P$ is a left $R$-module, ...
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Maximal ideals in a subring of the field of fractions [duplicate]

Let $R$ an integral domain with prime ideal $P.$ Let $$R_P=\{a/d:a,d\in R,d\not\in P\}.$$ We can show that $R_P$ is a subring of the field of fractions. The question is what are the maximal ideals of ...
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Surjective homomorphism from a faithfully flat module to a regular local ring.

Let $R$ be a regular local ring and let $M$ be a faithfully flat $R$-module. Does there necessarily exist a surjective $R$-module homomorphism from $M$ to $R$? For context, I am computing $\sum_{f\in\...
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1answer
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The vanishing ideal of the complex torus

I am trying to get my hands on the basics of algebraic geometry, and I am confused about the following. Let $(\mathbb{C}^{\times})^n \subset \mathbb{C}^n$ denote the complex torus. What I know is ...
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Intersections of Local Rings

Let $K$ be a field. Interestingly, the integrally closed subrings of $K$ are characterized as the intersections of valuation rings. What happens if we replace "valuation rings" with "local rings" ...
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Local ring if and only if $x$ or $1-x$ is unit

A commutative ring $R$ with identity is local if and only if for all $x \in R$, $x$ or $1-x$ or both is unit. I have solved the 'only if' part, but I have stuck on the 'if' part for a very long time$\...
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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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Completion of the local ring at a point on arithmetic surfaces.

Let $K$ be a number field and consider a arithmetic surface $X\to B=\operatorname{Spec} O_K$, i.e. $X$ is integral, regular, flat, proper over $O_K$ and it has dimension $2$. Now pick a closed point $...
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Consequence of local duality

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $d$ with a canonical module $\omega.$ Let $M,N$ be maximal Cohen-Macaulay $R$-modules. Then local duality implies $$\mathrm{Ext}^i(N,\...
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The residue field of valuations are finite extension

Let $K\mid k$ be a finitely generated extension (maybe of transcendence degree bigger than one) and consider a rank one valuation of $K$ over $k$, that is, a function $$v:K^\times\rightarrow \mathbb{R}...
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Is the ring $\mathbb{Z}_{p^n}$ local?

Let $p$ be a prime number. Is the ring $\mathbb Z_{p^n}$ a local ring? That is, the set of non-units an ideal of the ring? I think yes, because the only prime that divides the order of the ring is $p$...
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Every finite integral extension of a Henselian, pseudo-geometric and analytically normal ring is algebraically closed in its completion.

In the book Local Rings Nagata states in Theorem 44.1: If $R$ is a Henselian pseudo-geometric analytically normal ring, then every finite integral extension $R'$ of $R$ is analytically ...
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Theorem on Injective Dimension in Weibel's “An Introduction to Homological Algebra”

This fact is stated in Weibel's book without proof: If $A$ is a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak{m},k)$, then $id(A) \le d$ is equivalent to $\...
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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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Adjoining an element with given minimal polynomial to a DVR of characteristic p.

Let $A$ be a DVR of characteristic $p$, with $\pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(\alpha)$ where $\alpha$ has minimal polynomial $y^p+\...
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Showing some element is a zero divisor

Let $R=R_1\oplus R_2$, where each $R_i$ is a commutative ring with unity. Let $(S,\eta)$ be a local subring of $R$. Let $\pi_1$ be the projection of $R$ onto $R_1$. It is also given that $\pi_1|_S$ is ...
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Integral extension of a local ring is semilocal

Let $S\subseteq R$ be commutative rings with $1$. It is given that $S$ is local and $R$ is integral over $S$. I need to show that $R$ is semilocal that is $R$ has finitely many maximal ideals. It is ...
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Divided power algebra is artinian as a module over the polynomial ring

In the paper Homological algebra on a complete intersection, with an application to group representations I found the following argument that I do not understand: Suppose $B$ is a local artinian ring ...
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On cancelling special type of ideals in Noetherian local domain of dimension $1$

Let $(R,\mathfrak m)$ be a local, Noetherian domain of dimension $1$. If $J$ a non-zero ideal of $R$ such that $J^2=\mathfrak m J$, then is it true that $J^2=\mathfrak m^2$ ? If this is not true in ...
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On cancelling primary ideals in Noetherian, local, unique factorization domain of dimension $2$

Let $(R,\mathfrak m)$ be a local , Noetherian , UFD of dimension $2$. Let $J$ be an $\mathfrak m$-primary ideal ($\sqrt J=\mathfrak m$) of $R$ such that $J^2=\mathfrak m J$. Then, is it true that $J^...
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On a special type of ideal in local Artinian ring

Let $(R,\mathfrak m)$ be a local Artinian ring. If $J$ is a non-zero ideal of $R$ such that $J^2=\mathfrak mJ$, then is it true that $J=\mathfrak m$ ? or at least $J^2=\mathfrak m^2$ ? NOTE: $J^2=\...
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Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$

Let me paste the title: Is there a nonzero module $M$ over an artinian local ring $(R,\mathfrak m)$ such that $\mathfrak mM=M\ ?$ Of course such a module could not be finitely generated.
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On certain local, Noetherian , unique factorization domain whose maximal ideal is minimally generated by $3$ elements

Let $(R,\mathfrak m)$ be a Noetherian , local , UFD with $\mu(\mathfrak m)=3$ ( where $\mu(\mathfrak m):=\min\{|S| : \langle S\rangle=\mathfrak m\}$ ). Also assume that if $J$ is an ideal with $\sqrt ...
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How to find a regular parameter system of the local ring $ \mathbb{C} [X_0, \dots, X_n]_{(P \ )} $?

How to find a regular parameter system of the local ring : $ \mathbb{C} [X_0, \dots, X_n]_{(P \ )} $ with : $ (P \ ) $ a prime ideal of : $ \mathbb{C} [X_0, \dots, X_n] $ ( i.e : $ P \in \mathbb{C} [...
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Is a complete intersection ring, which is a quotient of a maximal $A$-sequence, Artinian?

Let $A$ be a noetherian regular local ring, $x_1,\dots,x_n$ a regular $A$-sequence and $B = A / (x_1,\dots,x_n)$. Then $B$ is a complete intersection ring by definition. If $(x_1,\dots,x_n)$ is a ...
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Quotients of powers of maximal ideal in a local ring.

I need a reference to a (hopefully) known result: I think I can quite easily prove that if $L$ is a commutative Noetherian local ring with maximal ideal $M$ then for any positive integer $n$, $L$-...
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Is a subring of a local ring still local?

If $B$ is a (unital) subring of a unital local ring $A$, is $B$ still a local ring? If not, under which assumption is it true?
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Multiplicity of point of algebraic plane curve

I'm reading Fulton's Curves at the moment, and I'm stuck on Theorem 2 of section 3.2. He has defined the multiplicity $m_P(C)$ of a point $P = (0,0)$ of curve $C = V(F)$ (let's say $C$ is irreducible ...
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Do we have a nice classification of injective modules over a local ring?

Let $R$ be a local ring. We know, for example, that $V$ is a projective $R$ module if and only if $V$ is free. Also, since $R$ is not semisimple, there must be at least one module which is not ...
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Corollary 4.19 from “Homological methods in commutative algebra”

I would like to show the following result: for a noetherian local ring $A$, we have $\mathrm{gl.dim}_A=\mathrm{hd}_A (k)$. Notice that the left side term of the equality is the global dimension of $A$,...
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1answer
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$R$ is artinian $\implies$ $(eR)_R$ is local $R$-module for an idempotent $e \in R$

Let $R$ be a ring with unity $1$ and artinian. Consider $R$ as right $R$-module $R_R$. Since it is artinian we have a finite decomposition of $R$ into right ideals: $$R_R = A_1 \oplus ... \oplus A_n$$...
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Example for being an integral domain is not a local property.

Let $K$ be a field and $R=K\times K$ the product ring. We know this ring is not integral domain. But for all $P \in Spec(R)$, $R_P$ (the localization of $R$ at $P$) is integral domain. I know this ...
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Calculate $\dim_k(\mathfrak{m}^i/ \mathfrak{m}^{i+1})$ for the local ring $\mathbb{Z}_{(2)}[\sqrt{-3}]$

I am just studying for my exam in commutative algebra and I am trying to compute the Hilbert polynomial of the ring $R=\mathbb{Z}_{(2)}[\sqrt{-3}]$. To do so, I was trying to calculate $\dim_k(\...
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Completion and endmorphism ring of injective envelope

Let $(R,m,k)$ be a commutative Noetherian local ring. Suppose $E$ is the injective envelope of $k$. For any module $M$, denote by $M^*=Hom_R(M,E)$. Denote by $R^~$the $m$-adic completion of $R$. Let $...
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1answer
102 views

Local ring of a surface: polynomial expression

Let $(X,\mathscr O_X)$ be a smooth surface over a field $k$ (two dimensional scheme, regular, noetherian....) and fix a point $x\in X$. Then if we complete the local ring $\mathscr O_{X,x}$ at its ...
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1answer
39 views

Artinian local ring with every non-maximal ideal being principal [closed]

Let $(R, \mathfrak m)$ be an Artinian local ring with every non-maximal ideal being principal. Then is $R$ a principal ideal ring ?
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1answer
64 views

Valuation ring of finite Krull dimension whose every non-maximal ideal is principal

Let $(R, \mathfrak m)$ be a Valuation ring of finite Krull dimension such that every non-maximal ideal i e. every ideal which is not $\mathfrak m$, is principal. Then is $R$ Noetherian i.e. a discrete ...
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28 views

Is R/S injective R-module?

In the book Exercises on Modules and Rings- Lam, Exercise 16.2 gives a ring as follows: K is a field, $\sigma\in End(K)$, $L=\sigma(K)$ such that $[K:L]=2$ (dimension is given as n in the book.) Let $...
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78 views

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 ?