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

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Is the ring $\mathbb{C}[t^2,t^3]$ integrally closed?

I am trying to understand if the ring $\mathbb{C}[t^2,t^3]$ in integrally closed (into its field of fractions), but I have no idea about how to proceed. All I have tried until now has failed. Any ...
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2answers
65 views

If $N\cap rM=rN$ for all $r\in R$, then is $M=N\oplus K$ for some $K$?

Suppose $M$ is a finitely generated free module over a principal ideal domain $R$, and $N$ a submodule. Why does the condition $N\cap rM=rN$ for all $r\in R$ implies that $M=N\oplus K$ for some ...
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0answers
77 views

Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left) noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by finite set $x_1,...,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. Hence a Poisson ...
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2answers
134 views

Dimension of quotient rings and zero divisors

In a previous question, I asked about the correctness of a method to compute the Krull dimension of quotient rings which works well if the ring in question is of the form $A/(x_1,\ldots,x_n)$, where ...
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1answer
32 views

Show that a subalgebra is commutative.

If $B$ is an unital algebra (even not commutative), how do I show that the subalgebra spanned by the elements $1$, $f$ and $(f - \lambda1)^{-1}$ is commutative? Thank you.
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1answer
85 views

Can a ring that is not finitely generated and contains $\mathbb{C}$ be Noetherian? [duplicate]

Suppose we have a ring R that contains the complex numbers, $\mathbb{C}\subset R$ and is not finitely generated as a ring. Can R be Noetherian?
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1answer
52 views

Kähler differentials of tensor product

Let $B,C$ be $A$-algebras. How can I show that $$ \Omega_{B \otimes_A C/A}=\Omega_{B/A}\otimes_A C \oplus \Omega_{C/A}\otimes_A B?$$
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1answer
68 views

Meaning of 'Isomorphism (with respect to inclusion)'

This is the first time that I see this phrase. I'm reading Commutative Algebra by N.Bourbaki. I'll extract 2 propositions that use this phrase. The first one is on page 68 of the book. ...
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1answer
58 views

Why is the tight closure tightly closed?

Let $R$ be a commutative noetherian ring containing a field of characteristic $p\gt0.$ For an ideal $I\subset R,$ the tight closure $I^*$ is defined as $$\{f\in R\mid \exists t\in R, ...
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1answer
69 views

Commutative rings with trivial automorphism group

The commutative rings $\mathbb{Z}/p\mathbb{Z}$, $\mathbb{Z}_{(p)}$, $\mathbb{Z}$, and $\mathbb{Q}$, where $p$ is a rational prime, all have trivial automorphism groups. Are there any other (unital) ...
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377 views

Krull dimension of quotient rings

This question is very related to this other question. I have an alternative solution to the ones proposed in the answers, and I'd like to know if it is correct. I want to find the dimension of ...
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1answer
51 views

If $X$ is affine reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$

If $\operatorname {Spec}A$ is reduced, show that $f\neq 0 \Rightarrow \overline {D(f)} = \operatorname {Supp} f$ Attempt at a solution: Clearly $D(f) \subset \operatorname{Supp} f$. Since the ...
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0answers
46 views

Show that $\operatorname{Spec}k[x_1,x_2,…,x_n]/(x_1^2+\cdots+x_m^2)$ is normal for $\operatorname{char}k\neq 2, n\ge m \ge 3$ [duplicate]

I want to show that if $F(T) \in B[T]$, where $B:=k[x_1,x_2,...,x_n]/(x_1^2+\cdots+x_m^2)$, is monic and has a root $\alpha \in\mathcal K(B)$ then $\alpha$ actually lives in $B$. This will imply that ...
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1answer
43 views

Gauss lemma in UFDs

Let $A$ be a UFD, and $f\in A$ a square-free element. Define the integral domain $B:=A[z]/(z^2-f)$, and consider a monic polynomial $F(T) \in B[T]$ such that $F(\alpha) = 0$ for some $\alpha \in ...
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1answer
110 views

Poincaré series of quotient module

I am trying to calculate the Poincaré series $P(M,t)$ with respect to the standard degree grading of the graded $\mathbb{C} [x,y,z,w]$-module $ M=\mathbb{C}[x,y,z,w]/I$, where $I = (x,w) \cap (z,w) ...
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1answer
212 views

Subrings of polynomial rings over the complex plane

I have the following questions: (i) must every subring of the polynomial ring in two variables over the complex plane, containing the complex plane itself, be Noetherian? (ii) Are there Noetherian ...
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1answer
89 views

Question about tensor product of homomorphisms

I've come to think about this problem when reading a proof in Commutative Algebra by N. Bourbaki. Say, let $R$ be a commutative ring, given 3 $R-$modules $A$, $B$, $C$, and the $R$-homomorphism $f:B ...
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2answers
196 views

power series ring is faithfully flat but not free

Let $A$ be a commutative ring. Question 1: Why is the power series ring $A[[x]]$ not free over $A$ in general? Question 2: Why is $A[[x]]$ faithfully flat over $A$?
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105 views

If $x$ is integral over $A_m$ for all maximal ideals $m$, then $x$ is integral over $A$

I am going over an old exam, and there is this question that I am stuck: Given $A$ a commutative ring with unity, show that if $x\in\operatorname{Frac}(A)$ is integral over $A_m$ for all maximal ...
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1answer
59 views

can the projective dimension be read from any projective resolution?

Let $P_{\bullet}, P'_{\bullet}$ be two projective resolutions of an $R$-module $M$. Denote their differentials by $d,d'$ respectively. Define $M_i = \operatorname{ker} d_{i-1}, M'_i = ...
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1answer
109 views

homotopy equivalence of projective resolutions

Let $P_{\bullet}$ and $P'_{\bullet}$ be projective resolutions of a module $M$ over a commutative ring $R$. Then $P_{\bullet}$ and $P'_{\bullet}$ are homotopy equivalent (see e.g. Matsumura, CRT, ...
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176 views

Finish a proof that every prime ideal of a ring is the contraction of a prime ideal in its formal power series

Given a commutative ring $A$ with identity, and its formal power series ring $A[[x]]$, I am attempting to prove that every prime ideal of $A$ is the contraction of a prime ideal of $A[[x]]$. ...
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1answer
65 views

Quotient of Jacobson ring is Jacobson as in Eisenbud

I wanted to prove the following: Let $R$ be a Jacobson ring, $\mathfrak{p}<R$ a (prime) ideal. Then $R/\mathfrak{p}$ is Jacobson. The statement has been taken from Eisenbud's Commutative ...
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369 views

Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...
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64 views

Is $A_{S}$ a ring of fractions or a localized ring?

Let $$A_{S}:=\Biggl\{\frac{a}{b}\in K\mid a\in A,b=1\vee b=st,\;\forall s,t\in S\Biggr\},$$ where $K=Frac(A)$, $A\subset K$, $A$-integral domain and $0\not\in S\subset A$. Is $A_{S}$ a ring of ...
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1answer
281 views

Atiyah Macdonald - 2.15 (direct limit)

Atiyah-Macdonald book constructs the direct limit of a directed system $(M_i,\mu_{ij})$, (where $i\in I$, a directed set, and $i\leq j$) of $A$-modules as the quotient $C/D$, where $C=\oplus_{i\in I} ...
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1answer
176 views

Computing the Hilbert-Poincaré series of a quotient

I am preparing for an exam of commutative algebra, and I am at loss about how to compute Hilbert-Poincaré series of rings. In particular, I have some preparation exercises I can't solve. Mainly they ...
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1answer
81 views

existence of a finite free resolution

Problem 2.1.26 in Bruns and Herzog, CMR, reads as follows: "Let $R$ be a Cohen-Macaulay local ring of dimension $d$ and $M$ a finite $R$-module. Deduce that the $d$-th syzygy of $M$ in an arbitrary ...
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1answer
79 views

Example where prime spectrum suits better than the maximal spectrum

in a lot of algebraic geometry books I've heard that working over $\mathrm{Spec}(A)$ is better than working over $\mathrm{Spmax}(A)$ in the case where you consider a variety over a non-agebraically ...
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3answers
394 views

An ideal whose radical is maximal is primary

I've got to prove that an ideal $Q$ whose radical is a maximal ideal is a primary ideal. That is, I want to prove that if $xy\in Q$, then $x\in Q$ or $y^n\in Q$ for some $n>0$. I've been ...
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1answer
84 views

On localization at a prime ideal

Let $(A,\mathfrak m)$ be a commutative ring with $\dim A=d$ and $\mathfrak p$ a prime ideal of $A$. If $(A/\mathfrak p)_{\mathfrak q}=0$ for all primes $\mathfrak q$ with ...
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1answer
178 views

On Gorenstein ring of dimension zero

Let $R$ be an Artinian local ring. Then $R$ is a Gorenstein ring (i.e., $R$ is an injective $R$-module) iff for any ideal $I$ of $R$, Ann$($Ann$(I))=I$. Why? (We call $R$ Gorenstein if injective ...
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162 views

Krull dimension and localization of a module

Let $(R,\mathfrak m)$ be a $d$-dimensional noetherian local ring, and $M$ an $R$-module. If $\mathfrak p$ is a prime ideal of $R$ with height $d-1$, then $\dim M_{\mathfrak p}=\dim M-1$?
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148 views

On rings with a unique maximal ideal

I would be grateful if you guide me through the following question: Suppose a commutative ring with identity, $R$, has a unique maximal ideal, say $M$. If $M$ is principal, can we show that every ...
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1answer
77 views

Flatness question

In reading on the stacks project I came across a result I don't quite follow: "Assume M is finitely presented and flat, i.e., (1) holds. We will prove that (7) holds. Pick any prime p and x1,…,xr∈M ...
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1answer
79 views

Question on Nakayama?

In reading a certain proof on the stacks project "http://stacks.math.columbia.edu/tag/00NV", I can't see how Nakayama's lemma is used to make the following conclusion: "Assume M is finitely presented ...
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1answer
39 views

proving the existence of a regular element in the quotient of a completion

Let $(R,m)$ be a local Cohen-Macaulay ring and $p$ a prime ideal of $R$. Denote by $\hat{R}$ the completion of $R$ with respect to the $m$-adic topology. Take $q \in \operatorname{Ass}(\hat{R}/p ...
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1answer
126 views

Difference between graded ring and graded algebra

Wikipedia says that a graded $A$-algebra is just a graded $A$-module that is also a graded ring. Question: when one says then "finitely generated graded $A$-algebra", does one mean that every element ...
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1answer
83 views

tensor product of $R$-module homomorphisms

Let $M,M',N,N'$ be modules over a commutative ring $R$, and $f:M\to M'$ and $g:N\to N'$ are $R$-modules homomorphisms. Then prove or disprove the following statements. a) If $f$ and $g$ are ...
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47 views

Proof of Proposition 2.1.1 in Bruns and Herzog

Let $k$ be a field, $R$ a $k$-algebra and $K$ an extension field of $k$ that is finitely generated over $k$. Then there exists a chain of cyclic extension fields $k=K_0 \subset K_1 \subset \cdots ...
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1answer
71 views

Is $k[x,y]/(x,y) \cong k$?

Let $k$ be an algebraically closed field and consider the $k[x,y]$-module $k[x,y]/(x,y)$. By the Nullstellensatz, $(x,y)$ is a maximal ideal of $k[x,y]$, hence $k[x,y]/(x,y)$ is a field. From this, I ...
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1answer
56 views

Why is $M=\{(x_i)\in R^n: \sum r_ix_i=0\}$ a projective module?

Suppose $R$ is a commutative unital ring with generators $r_1,\dots, r_n$. How can we see that the submodule $$ M=\{(x_1,\dots,x_n)\in R^n:\sum_{i=1}^n r_ix_i=0\} $$ is a projective submodule? I ...
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2answers
131 views

When do we have $\operatorname{depth}_{A} B = \operatorname{depth}_B B$?

Let $(A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a local homomorphism with $A$ a regular local ring. Assume further that this ring map is finite. How can we prove that $\operatorname{depth}_B B = ...
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1answer
89 views

Grade of the Hom functor

Let $R$ be a Noetherian ring, $I$ an ideal and $M, N$ finite $R$-modules. Prove that $\operatorname{grade}(I,\operatorname{Hom}_R(M,N))\ge\min(2,\operatorname{grade}(I,N))$. This question is ...
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41 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
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1answer
239 views

Separability and tensor product of fields

Is it true that a finite degree field extension $L/k$ is separable if and only if $L\otimes_{k}L$ is a reduced $L$-algebra? Surely the "only if" part is true because if the extension is ...
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1answer
193 views

A problem about localization of $\mathbb{Z}/6\mathbb{Z}$ at prime ideal $2\mathbb{Z}/6\mathbb{Z}$

We know that Given a prime ideal $P$ of a commutative ring, there is a one-to-one correspondence between $\lbrace\text{prime ideals }Q\subset P\rbrace$ and $\lbrace\text{prime ideals of } S^{-1}R ...
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2answers
77 views

does a prime in an extensions of integral domains remain radical?

Let $R\subset R'$ be an extension of integral domains. So we have an inclusion map $i:R\hookrightarrow R'$. Let $\mathfrak{p}\subset R$ be a prime ideal. We know that $\mathfrak{p}^e$ (generated by ...
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2answers
85 views

Question on the dimension $\dim 0$.

What is the Krull dimension $\dim 0$ of the trivial ring? Trivial ring is denoted by $0$