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

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10
votes
3answers
353 views

Is the coordinate ring of SL2 a UFD?

Is the ring $K[a,b,c,d]/(ad-bc-1)$ a unique factorization domain? I think this is a regular ring, so all of its localizations are UFDs by the Auslander–Buchsbaum theorem. However, I know there ...
2
votes
1answer
51 views

Why is $B[x]/M$ algebraic over $B/m$?

Let $B$ be a subring of some field $K$, $x$ some element in $K$, $m$ a maximal ideal in $B$ and $m[x]$ the extension of $m$ in $B[x]$ and $M$ a maximal ideal in $B[x]$ such that $m[x] \subset M $ and ...
3
votes
2answers
348 views

Question about integral closures and localizations

Suppose $A$ is an integral domain with integral closure $\overline{A}$ (inside its fraction field), $\mathfrak{q}$ is a prime ideal of $A$, and $\mathfrak{P}_1,\ldots,\mathfrak{P}_k$ are the prime ...
3
votes
1answer
603 views

Proof of Hilbert's Nullstellensatz

I'm working through my notes and I'm stuck in the middle of the proof of Hilbert's Nullstellensatz. (Hilbert's Nullstellensatz) Let $k$ be an algebraically closed field. Let $J$ be an ideal in ...
4
votes
2answers
444 views

A question about a proof of a weak form of Hilbert's Nullstellensatz

I'm trying to prove the following (corollary 5.24 page 67 in Atiyah-Macdonald): Let $k$ be a field and let $B$ be a field that is a finitely generated $k$-algebra, i.e. there is a ring homomorphism ...
3
votes
2answers
495 views

A question about a weak form of Hilbert's Nullstellensatz

Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows: Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$. We know ...
5
votes
1answer
494 views

Direct limit of rings

I have a (perhaps silly) question motivated by the following exercise in Atiyah-MacDonald's book: Say $(A_i,\alpha_{ij})$ is a direct system of rings with direct limit $A_\infty$ and associated ...
7
votes
0answers
166 views

Question about definition of $\mathrm{Ext}$

One can define $\mathrm{Ext}^n(M,N)$ (where $M,N$ are $R$-modules) in two ways, either by taking an injective resolution of $N$ and applying $\mathrm{Hom}(M,-)$or by taking a projective resolution and ...
2
votes
1answer
252 views

Compute Hilbert function of a monomial ideal

I'd like to know whether there exist easy methods that compute the Hilbert function of a graded $k$-algebra, without computer programs. My homework asks to me to compute the Hilbert function of ...
0
votes
3answers
220 views

$x$ algebraic over $K$, $v$ a polynomial in $x$ then $v$ algebraic?

In the proof of proposition 5.23 Atiyah-Macdonald on page 66 use that if $x$ is algebraic over $K$ and $v = a_n x^n + \dots + a_1 x + a_0$ then $v$ is algebraic over $K$ (where $K$ is the field of ...
12
votes
3answers
679 views

What are rational integer coefficients?

I have a question about the following excerpt from Atiyah-Macdonald (page 30): “A ring $A$ is said to be finitely generated if it is finitely generated as a $\mathbb Z$-algebra. This means ...
1
vote
2answers
148 views

Another question about a proof in Atiyah-Macdonald

I have a question about the following proof in Atiyah-Macdonald: 1:Why is $\Omega$ infinite? Are all algebraically closed fields infinite? 2: How does the existence of $\xi$ follow from $\Omega$ ...
1
vote
2answers
541 views

Example of non-finitely generated $R$-algebra

By definition, an $R$-algebra is a ring homomorphism $f: R \to S$. For example, if $R=\mathbb Z$ and $S= \mathbb Z / n \mathbb Z$ then the projection $k \mapsto k \mod n$ is a ring homomorphism so ...
5
votes
1answer
160 views

Possibly false proof in AM

Here is the excerpt of the book where I suspect a mistake (page 66): Where they say "The restriction to $A$ of the natural homomorphism $A^\prime \to k^\prime$" I think we don't want a restriction. ...
2
votes
0answers
285 views

Completion and Tensor Product of Algebras

Let $A$ be a commutative ring with 1, $I$ an ideal in $A$, $B$ an $A$-algebra. I am trying to prove the following isomorphism of $A$-algebras: $$ \big( A^* \otimes _A B \big) ^* \cong B^* $$ "$^*$" ...
1
vote
1answer
65 views

$(B/m)[x] = B[x]/M$?

Assume $K$ is a field and $B$ is a subring of $K$ and $x \in K$. Let $m$ be a maximal ideal of $B$. Let $m^e$ denote the extension of $m$ in $B[x]$. Let $M$ be a maximal ideal in $B[x]$ containing ...
2
votes
0answers
252 views

Noetherian condition on the ring of formal power series without Axiom of Choice

I use the definition of a Noetherian ring given by Qiaochu in this: A commutative ring is Noetherian if, for any nonempty collection of ideals $\mathcal{I}$, there is some $I \in \mathcal{I}$ which is ...
2
votes
0answers
89 views

How to tell if an ideal is absolutely prime

Consider the ideal $I=(ag-ec-1,ah+bg-cf-de)$ of $R=K[a,b,c,d,e,f,g,h]$. Is $I$ prime when $K=\overline{\mathbb{F}}_2$ is the algebraic closure of a field of 2 elements? Can computers answer this ...
2
votes
0answers
144 views

A question about the proof that for an integral domain $D \subset K$ there is a valuation ring

I have some questions about a proof in Atiyah-Macdonald. It starts on page 65 where they write "...We want to prove that $B$ is a valuation ring of $K$." Let me modify the claim a tiny bit and ...
7
votes
1answer
250 views

Question about proof of Going-down theorem

I have written a proof of the Going-down theorem that doesn't use some of the assumptions so it's false but I can't find the mistake. Can you tell where it's wrong? *Going-down*$^\prime$: Let $R,S$ ...
4
votes
1answer
311 views

Existence of valuation rings in a finite extension of the field of fractions of a weakly Artinian domain without Axiom of Choice

Can we prove the following theorem without Axiom of Choice? This is a generalization of this problem. Theorem Let $A$ be a weakly Artinian domain. Let $K$ be the field of fractions of $A$. Let $L$ ...
5
votes
2answers
284 views

$F, G \in k[X_1, \dots , X_n]$ homogeneous of degrees $r$ and $r+1$ $\implies$ $F+G$ is irreducible

I have a question about Exercise 2-34 from William Fulton's Algebraic Curves book. The exercise is as follows. Suppose that $F, G \in k[X_1, \dots , X_n]$ are forms (i.e. homogeneous ...
6
votes
1answer
660 views

The Frobenius endomorphism

Let $\mathbf F$ be a field of prime characteristic $p$. It is known that the Frobenius map $c\phi=c^p~~\forall c\in\mathbf F$ is an endomorphism of $\mathbf F$. Moreover, since the only ideals of ...
1
vote
1answer
217 views

Dedekind's theorem on an integrally closed algebra over a commutative ring without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. Can we prove the following theorem without Axiom of Choice? If the answer is affirmative, by using ...
4
votes
1answer
451 views

Krull-Akizuki theorem without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. We use the definitions in my answers to this question. Can we prove the following theorem without ...
1
vote
1answer
396 views

Dedekind's theorem on a weakly Artinian integrally closed domain without Axiom of Choice

Let $A$ be a commutative ring. Let $f$ be any non-zero element of $A$. Suppose that $A/fA$ has a composition series as an $A$-module. Then we say $A$ is a weakly Artinian ring (this may not be a ...
5
votes
0answers
329 views

Ideals generated by regular sequences are generated by regular sequences in any order/Eisenbud, Exercise 17.6

I have a question regarding an exercise I found in Eisenbud's Commutative Algebra with a view towards Algebraic Geometry: Exercise 17.6: Any ideal of a Noetherian ring generated by a regular ...
4
votes
2answers
168 views

Insight of some concepts in commutative algebra

I really enjoyed the basic algebra course and wanted to teach myself a little more. So I am trying to learn commutative algebra from Atiyah-MacDonald and Eisenbud. The department in our university ...
4
votes
2answers
999 views

Nilpotency of the Jacobson radical of an Artinian ring without Axiom of Choice

Let $A$ be a commutative ring. Suppose $A$ has a composition series as an $A$-module. EDIT Since $A$ has a composition series, $A$ has a maximal ideal. Let $J$ be the intersection of all the maximal ...
-4
votes
2answers
496 views

Dedekind's theorem on an integrally closed algebra over a field without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. Can we prove the following theorem without Axiom of Choice? Theorem Let $A$ be an integrally ...
5
votes
1answer
363 views

Hilbert's Nullstellensatz without Axiom of Choice

Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951. Can we prove the following theorem without Axiom of Choice? Theorem Let $A$ be a commutative ...
22
votes
3answers
442 views

$\operatorname{Ann}(M\otimes_A N)=\operatorname{Ann}M+\operatorname{Ann}N$?

In the course of working on an exercise in Atiyah-MacDonald (exercise 3 on p. 31), I've come to the belief that, for $A$ an arbitrary commutative ring and $M,N$ arbitrary $A$-modules, ...
7
votes
1answer
576 views

Universal property of the completion of rings / modules

If $A$ is a noetherian local ring and $M$ an $A$-module, then we define the completion $\hat{M}$ of $M$ with respect to the stable $\mathfrak{m}$-filtration $\{M_n\}$ by ...
1
vote
3answers
353 views

Existence of a prime ideal in an integral domain of finite type over a field without Axiom of Choice

Let $A$ be an integral domain which is finitely generated over a field $k$. Let $f \neq 0$ be a non-invertible element of $A$. Can one prove that there exists a prime ideal of $A$ containing $f$ ...
1
vote
0answers
83 views

Two questions concerning integral dependence

Proposition 2.4 in Janusz's Algebraic Number Fields states that if $R$ is an integral domain with quotient field $K$, $L/K$ a field extension and $b \in L$ algebraic over $K$ with minimal polynomial ...
4
votes
1answer
263 views

The completion of a noetherian local ring is a complete local ring

We have defined the completion of a noetherian local ring $A$ to be $$\hat{A}=\left\{(a_1,a_2,\ldots)\in\prod_{i=1}^\infty A/\mathfrak{m}^i:a_j\equiv a_i\bmod{\mathfrak{m}^i} \,\,\forall ...
2
votes
2answers
48 views

Curve has a point which is either singular or has a tangent line parallel to the y-axis.

Suppose $k$ is an algebraically closed field with characteristic 0. Suppose $f(x,y)\in k[x,y]$ is irreducible and viewing $f(x,y)$ as a polynomial over $k[x]$ which is monic in $y$ and of degree>1 in ...
4
votes
1answer
242 views

Question about proof of Going-down theorem and prop. 3.16 in AM

Prop. 3.16 tells us that if $f: A \to B$ is a ring homomorphism and $\mathfrak p$ is a prime ideal of $A$ then $\mathfrak p$ is the contraction of a prime ideal of $B$ if and only if $\mathfrak p^{ec} ...
1
vote
2answers
109 views

Quibble with terminology

Proposition 5.15 on page 63 in Atiyah-Macdonald goes as follows: Let $A \subset B$ be integral domains, $A$ integrally closed, and let $x \in B$ be integral over an ideal $ \mathfrak a$ of $A$. Then ...
13
votes
4answers
2k views

An integral domain whose every prime ideal is principal is a PID

Does anyone has a simple proof of the following fact: An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
7
votes
2answers
713 views

Lying-over theorem without Axiom of Choice

This question is motivated by this and this. Can the following proposition be proved without Axiom of Choice? Proposition: Let $k$ be a field. Let $A$ and $B$ be commutative algebras without ...
5
votes
1answer
316 views

Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?

This question is related to the one here: A question related to krull akizuki In the answers to that question, some examples are given of a discrete valuation ring $A$ and a finite (necessarily ...
9
votes
0answers
177 views

Tensoring is thought as both restricting and extending?

I hope these questions are not too trivial. Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring $$ (R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c ...
2
votes
0answers
234 views

Proof of going-up theorem

Can you tell me if my proof is correct? Thanks. (I'm using propositions 5.6 and 5.10 from Atiyah-Macdonald which I proved separately.) Theorem: Let $R$ be integral over $S$. Let $p_1 \subset p_2$ be ...
3
votes
1answer
162 views

Why is a strict $p$-ring whose residue ring is a field necessarily local?

Let $A$ be a strict $p$-ring. Recall that this means $A$ is $p$-adically separated and complete, $p:A\rightarrow A$ is injective, and $A/pA$ is a perfect $\mathbf{F}_p$-algebra. If $A/pA$ is a field, ...
2
votes
0answers
112 views

What is the notation behind the $k[x_1,\ldots,x_n]$?

I don't understand the proof of Noether Normalization Lemma in "Algebraic Geometry and Arithmetic Curves" . Liu considers first the case $k[X_1,X_0]/I$, then $k[X_1,X_1]/I=k[X_1]/I$, then again ...
4
votes
1answer
152 views

If the special fiber of a flat morphism is reduced, then any other fiber is reduced?

Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$. If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, ...
1
vote
1answer
120 views

When Can I Conclude Two Algebras are Isomorphic?

First, $R$ is a commutative ring with unit. I have two $R$-algebras, $A$ and $B$. I have an isomorphism $$ A\otimes_R\mathbb{Z}\longrightarrow B\otimes_R\mathbb{Z}. $$ Is there a theorem that says ...
5
votes
1answer
368 views

trivial Picard group

let $S=\operatorname{Spec}(A)$ be an affine scheme. For which ring $A$, not field is it known that $H^1(S,\mathcal{O}_S^{*})$ is trivial? If $X\to S$ is a finite map and $H^1(S,\mathcal{O}_S^{*})$ is ...
3
votes
2answers
58 views

Proof of $R/I$ integral over $S/(S \cap I)$

Can you tell me if my reasoning is correct? I want to prove if $S \subset R$ are rings and $R$ is integral over $S$ and $I$ is an ideal of $R$ then $R/I$ is integral over $S/ (S\cap I)$. Let $R$ be ...