Questions about commutative rings, their ideals, and their modules.

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281 views

Residue field of a local ring as field extension

Let $k$ be a field, $A$ a finitely generated commutative $k$-algebra and $\mathfrak p$ a prime ideal of $A$. Let $K$ be the residue field of the local ring $A_\mathfrak{p}$. I want to show that $K$ is ...
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1answer
114 views

Question about completion of DVR.

Let $(R, (\pi))$ be a discrete valuation ring with residue class field $R/(\pi) \cong k$. It is well known that if $k$ embedds into $R$, then there is an isomorphism of the completion $\hat{R} \cong k ...
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1answer
66 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the prime ...
3
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1answer
318 views

Exercise from Kaplansky's Commutative Rings and Eakin-Nagata Theorem

Exercise 15 of section 2-1 of Kaplansky's Commutative Rings is to show that if $T$ is a Noetherian ring and is finitely generated module over a subring $R$ of $T$, then $R$ is Noetherian. Kaplansky ...
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1answer
72 views

Possible examples where the Zero Divisor Conjecture does not hold

Given a ring $R$ with a nonzero zero divisor $x$, it is easy to show that if $M$ is a nonzero $R$-module, then there exists $y\in R-\{0\}$ such that $ym=0$ for some $m\in M-\{0\}$. I was ...
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1answer
109 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
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1answer
69 views

For some finitely many nonzero prime ideals, the contraction and extension of their product is zero

I was reading P.M. Eakin's thesis paper, The converse to a well known theorem on Noetherian Rings. The following is taken from Theorem 2, page 281 of that paper, and that's where I'm stuck. Let $...
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1answer
116 views

Prime ideal is contraction of prime ideal iff it's saturated

Let $\varphi: A\to B$ be a commutative ring homomorphism and $P$ a prime ideal of $A$. The expansion of an ideal $I\subset A$ is the ideal generated by $\varphi(I)$ in $B$, and the contraction of an ...
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1answer
90 views

2-dimensional Cohen-Macaulay domain

I am searching for a $2$-dimensional Cohen-Macaulay (normal or not) domain. Thanks in advance for any suggestion.
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1answer
183 views

Is this module noetherian?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Assume that $A$ is a noetherian ring, and let $I\subseteq A$ be a proper ideal. Consider the ideal $I\otimes_k A \subseteq A\otimes_k A$....
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1answer
71 views

A quotient of a regular local ring may not be regular

Let $(R,m)$ be a regular local ring having an ideal $I$ such that $I$ is a subset of $m^2$. If $I$ possesses a non-zerodivisor, I want to show that $R/I$ can not be regular. My try is just that $m$ ...
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1answer
133 views

Maximal linearly independent sets in a f.g. module

Suppose $M$ is a finitely generated module over a commutative unital ring $R$. Is it true that every maximal linearly independent set in $M$ has the same size? What is the most general condition ...
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1answer
106 views

Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual. [closed]

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose $D(M)=\operatorname{coker}\...
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1answer
203 views

A finite module over a Noetherian ring is torsionless if and only if it is a submodule of a finite free module

Let $R$ be a Noetherian ring, and $M$ a finite $R$-module. Then $M$ is torsionless if and only if it is a submodule of a finite free module, where torsionless is defined here. (Bruns and Herzog, ...
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2answers
1k views

What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
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2answers
128 views

If $\{M_i\}_{i \in I}$ is a family of $R$-modules free, then the product $\prod_{i \in I}M_i$ is free?

If $\{M_i\}_{i \in I}$ is a family of free $R$-modules, then $\bigoplus_{i \in I}M_i$ is free. Is this true for the product $\prod_{i \in I}M_i$ too?
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1answer
106 views

Conditions for a quotient module to be Noetherian

I'm solving this problem from "Introduction to Commutative Algebra" of Atiyah and Macdonald. Here is the problem: Let $M$ be an $A$-module and let $N_1, N_2$ be submodules of $M$. If $M/N_1, M/N_2$...
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1answer
93 views

Depth zero module and $R$-regular element

Let $(R,m)$ be a commutative Noetherian local ring with $\operatorname{depth}(R)>0$ and $M$ be a finitely generated $R$-module with $\operatorname{depth}(M)=0$. Then can we take an $R$-regular ...
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1answer
71 views

Another non-regular C-M ring

Among other examples of Cohen-macaulay rings which are not regular I am run into $R=F[x]/(x^2)$ with $F$ a field. It is clear that it is not regular, since every regular local ring must be a domain ...
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1answer
134 views

An example of an $m$-primary ideal in noetherian local domain

Is there any example of a $m$-primary ideal $I$ in a noetherian local domain $(R, m)$ such that $I^2=mI\not=m^2 $?
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1answer
87 views

Both $R$ and $R/I$ are regular local rings

Let $R$ be a Noetherian local ring and $I$ is an ideal of $R$ such that both $R$ and $R/I$ are regular local rings. Could we deduce that $I$ is generated by an $R$-sequence? I know that a noetherian ...
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1answer
87 views

Issue in the first French edition of Serre's local fields

I've been reading Serre's Corps Locaux, and I believe my copy is a first edition, as there's only one copyright date listed, 1968. I believe I found an issue on page 57, which (if you're looking at ...
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1answer
100 views

Generalisation of a result on Kahler differentials

Let $B$ be a local ring which contains a field $k$ of characteristic zero, isomorphic to its residue field $B/\mathfrak{m}$. We know that the map $\delta:\mathfrak{m}/\mathfrak{m}^2 \to \Omega^1_{B/k} ...
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1answer
243 views

Modules over local artinian rings

What is known about the structure of finitely generated modules over local artinian commutative rings $R$? Any information is appreciated. Let us denote by $\mathfrak{m}$ the maximal ideal and by $k$ ...
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2answers
134 views

Can zero divisors be in the denominator when we localize rings?

Can we localize rings with zero divisors? Can those zero divisors be in the denominator? I thought defining $$\frac{a}{b}=\frac{c}{d} \text{ iff }t(ad-bc)=0 \text{ where $b,d,t$ belong to the same ...
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0answers
135 views

Local complete intersection scheme, conormal sheaves and differentials

Let $X$ be a smooth projective variety over $\mathbb{C}$ and $Z \subset X$ be a local complete intersection subscheme in $X$. Denote by $I_Z$ the ideal sheaf of $Z$ in $X$ and $\Omega^1_X$ the sheaf ...
3
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1answer
109 views

Isomorphism between Ext groups in Huybrechts and Lehn's book Geometry of Moduli Spaces of Sheaves

On p.46 (or p. 43 in the 1st edition) of Huybrechts and Lehn book Geometry of Moduli Spaces of Sheaves, 2nd ed., they write: Since $K$ is $A$-flat and $I \otimes_k F_0$ is annilated by $m_A$, ...
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1answer
88 views

Hilbert Nullstellensatz and ring of continuous functions

Is there any relation between Hilbert's Nullstellensatz and the fact that the maximal ideals in $\mathcal C([0,1])$ correspond to a point in $[0,1]$ (which can be generalized to compact hausdorff ...
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1answer
51 views

What is $\overline{Y}$ in $\text{Spec}A$?

Consider a subset $Y$ of $\text{Spec}(A)$. (Here $A$ is a commutative ring.) What is the closure of $Y$ (or $\overline{Y}$)? I have been under the impression that $\overline{Y}$ is the set of prime ...
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1answer
90 views

If $\mathbb{Z}$ satisfies an identity $\eta$, then every **commutative** ring satisfies $\eta$? And related questions.

Assume all rings have unity and that ring homomorphisms preserve unity. Now by general principles, if every free object in the category of rings satisfies an identity $\eta$, then every object in the ...
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1answer
94 views

If a proper ideal contains some power of a maximal ideal then the maximal ideal is the only prime ideal that contains the ideal.

Let $R$ be a commutative ring with $1$ and $\mathfrak{m}\subset R$ be a maximal ideal. Show that if $I\subset R$ is a proper ideal containing $\mathfrak{m}^n$ for some $n\geq 1$, then $\mathfrak{m}$ ...
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1answer
96 views

Use of Zorn's Lemma in showing nilradical equals intersection of primes

I'm very confused as to the use of Zorn's lemma in showing that the nilradical of a ring is the intersection of all the prime ideals. Namely, we let $a \notin N$, where $N$ is the nilradical. Then we ...
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1answer
135 views

Reid, Undergraduate Commutative Algebra, Exercise 0.23

Let $f \in A$; if $f$ is reducible then the principal ideal $(f)$ is contained in a bigger principal ideal $(f_1)$. Consider the following conditions on a ring $A$. (a) $A$ is a UFD; (b) every ...
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0answers
214 views

Proof that $\mathbb{Z}[\sqrt{-5}]$ is integrally closed

There are demonstrations on the Internet saying that the polynomial $$\left(x-\frac{a}{c}-\frac{b}{d}\sqrt{-5}\right)\left(x-\frac{a}{c}+\frac{b}{d}\sqrt{-5}\right)$$ is monic if and only if $\...
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1answer
47 views

an apparent contradiction regarding the local ring at a point

I have encountered an apparent contradiction: Let $Y$ be an affine variety of $\mathbb{A}^n$ and $P$ a point of $Y$. Then i have proved that $\mathcal{O}_P$ is an integral domain and it is also not an ...
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1answer
178 views

Localization of a direct product

Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
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1answer
82 views

Grade of maximal ideals in polynomial rings over Artinian local rings

If $R$ is a commutative Artinian ring it is well-known that $R$ is Cohen-Macaulay. Also, if $S$ is a Cohen-Macaulay ring, then any polynomial ring $S[X_1,\dots,X_n]$ is so. Now if $R$ is a commutative ...
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71 views

Can $R[[x]]$ contain constants?

Consider the ring $R[[x]]$ of formal power series $\sum_{n=0}^\infty a_nx^n$ with coefficients in $R$. I was wondering whether $R[[x]]$ contains elements of $R$ (polynomials of degree $0$). I'm ...
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1answer
102 views

Is $R[X]/(f)$ Cohen-Macaulay if $R$ is so?

Let $R$ be a commutative (Noetherian) Cohen-Macaulay ring, and $f \in R[X]$ be monic. I guess that $R[X]/(f)$ is also Cohen-Macaulay. Is my hunch valid? Thanks for any help.
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0answers
116 views

Flat base change preserves the property of being non-degenerate

We say a homomorphism $f:A\rightarrow B$ of noetherian rings is non-degenerate if the induced map $f^*:{\rm Spec}(B) \rightarrow {\rm Spec}(A)$ maps ${\rm Ass}(B)$ into ${\rm Ass}(A)$. Let $f:A \...
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1answer
70 views

Let $A= \mathbb{Z}[x]$ and $ m=(2,x)$. Find the Krull dimension of $A_m$.

Let $A=\mathbb{Z}[x]$ and $m=(2,x)$. $1$. Then what is the Krull dimension of $A_m$? $2$. If $B=A_m/(x^2-125)$ what is the Krull dimension of $ B $? Any suggestions?
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99 views

About weakly associated primes

Let $A$ be a commutative ring, and $M$ an $A$-module. A prime ideal $\mathfrak{p}\subset A$ is said to be weakly associated to $M$ if it is minimal over some $\operatorname{ann}m$, where $m\in M$. I ...
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1answer
137 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for $S_n$ ...
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0answers
51 views

Free resolution by Groebner basis

I am studying approaches of Groebner basis in Homological and commutative algebra. I am so confused how can I find the minimal resolution for the below ideal $$I=\langle yz-xw,y^3-x^2z,xz^2-y^...
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0answers
30 views

Why is $A/I^n A \cong \hat{A}/I^n \hat{A}$?

Let $A$ be a commutative ring, $I \subset A$ a finitely generated ideal. Define $\hat{A} := \varprojlim A/I^n$. What is the best way to proof that $A/I^nA \cong \hat{A}/I^n \hat{A}$ for all $n$?
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59 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
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2answers
210 views

A quotient of a Cohen-Macaulay ring

Let $S$ be a Cohen-Macaulay local ring, $I$ an ideal of $S$ and $R=S/I$. If we know that $I$ is generated by $\dim S-\dim R$ elements could we infer that $R$ is Cohen-Macaulay? Thanks in advance!
4
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1answer
230 views

Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
4
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3answers
96 views

ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?

I am following an argument in chapter zero of Eisenbud's Commutative Algebra book. It is not clear whether or not he is assuming that $R$ is a domain. If I start the proof assuming $R$ is not ...
4
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1answer
185 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ (...