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

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3
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2answers
54 views

Example needed: R non-graded Gorenstein and gr(R) is also Gorenstein Artin k-algebra.

I am wondering if there is a non-graded Gorenstein Artin k-algebra R such that its associated graded ring, gr(R), is also Gorenstein. All the non-graded Gorenstein rings I tried so far have not ...
4
votes
1answer
1k views

A good commutative algebra book [duplicate]

Possible Duplicate: Reference request: introduction to commutative algebra I'm looking for a good book on commutative algebra covering most of (but not limited to) : Basic Galois theory ...
4
votes
1answer
68 views

If $R_2$ is an $R_1$-algebra, then is $R_2 \otimes_{R_1} M$ an $R_2$-module?

If we have a ring homomorphism $f\colon R_{1}\rightarrow R_{2}$, and if $M$ is an $R_{1}$-module, my question is: Can we show that the $R_{1}$-module $R_{2}\otimes_{R_{1}}M$ is somehow also an ...
4
votes
3answers
360 views

Height one prime ideal of arithmetical rank greater than 1

Let $R$ be a Noetherian local domain which is not a UFD and let $P$ be a height one prime ideal of $R.$ Can we find an element $x\in P$ such that $P$ is the only minimal prime ideal containing $x$?
3
votes
3answers
207 views

Does localization preserves dimension?

Does localization preserves dimension? Here's the problem: Let $C=V(y-x^3)$ and $D=V(y)$ curves in $\mathbb{A}^{2}$. I want to compute the intersection multiplicity of $C$ and $D$ at the origin. ...
2
votes
0answers
220 views

Localization of an infinite product of fields at its maximal ideals

Let $k$ be a field. Let $R = \prod_{n \in \mathbb{N}} k$. Due to the answer in this question Infinite product of fields, we know that $R$ is zero dimensional, and the localization $R_m$ at every ...
5
votes
3answers
1k views

Ideal of the twisted cubic

The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
22
votes
1answer
684 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
3
votes
1answer
230 views

Quotient of a Regular local ring.

Is the quotient of a regular local ring by a prime ideal Cohen-Macaulay? If so, how can we see this, if not, is there a counterexample? We know that a regular local ring is a UFD, so $0$ is a prime ...
1
vote
1answer
308 views

A lemma on the integral closure of a Noetherian domain of dimension 1

I need to prove the following lemma(?) which is motivated by this and this. Lemma Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral ...
3
votes
3answers
176 views

Well definition of multiplicity of a root in a polynomial of ring.

Let $R$ be a conmutative ring with identity, let $\displaystyle f=\sum_{k=0}^{n}a_k x^k \in R[x]$ and $r\in R$. If $f=(x-r)^m g$, $m\in\mathbb{N}$ and $g\in R[x]$ with $g(r)\neq 0$, then the root $r$ ...
4
votes
3answers
1k views

Direct proof that, in a commutative ring with only one prime ideal $P$, every element of $P$ is nilpotent

Let $R$ be a commutative ring with identity such that $R$ has exactly one prime ideal $P$. Prove: all elements in $P$ are nilpotent. While doing this problem, I used the fact that "the ...
2
votes
1answer
317 views

Homomorphisms into complex numbers

Let $I$ be a set of huge cardinality, that is, let $|I|>\mathfrak{c}$. Consider the real product algebra $\mathbb{R}^I$ of all real functions defined on $I$. Can we determine: 1) all algebra ...
2
votes
0answers
578 views

The group of invertible fractional ideals of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $I(A)$ be the group of invertible fractional ideals of $A$. Let $P$ be a maximal ideal of $A$. Let $I(A_P)$ be the group of invertible fractional ...
3
votes
1answer
115 views

Difference between $\left< x\right> \cap \left< x,y\right>^2$ and $\left< x,y\right>^3$

Consider the ideals $I = \left< x\right> \cap \left< x,y\right>^2 = \left<x^3,x^2y, xy^2\right>$ and $J=\left< x,y\right>^3=\left< x^3, x^2y, xy^2, y^3 \right>$ in ...
3
votes
1answer
276 views

Completion of regular local rings

Let $K$ be a complete field with respect to a discrete valuation $v$ and let $O_K$ be its valuation ring, $m$ its maximal ideal. Suppose $K$ has characteristic $0$ and that $O_K/m$ is of ...
2
votes
1answer
186 views

Bruns-Herzog Problem 2.1.26, page 64

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 finite free resolution is either $0$ or a maximal ...
8
votes
3answers
416 views

Commutative Algebra without the axiom of choice

It is well known that in a commutative ring with unit, every proper ideal is contained in a maximal ideal. The proof uses the axiom of choice. This fact, and others that are proved using essentially ...
6
votes
2answers
497 views

Motivation for Koszul complex

Koszul complex is important for homological theory of commutative rings. However, it's hard to guess where it came from. What was the motivation for Koszul complex?
3
votes
1answer
63 views

Coefficients in products and powers of large polynomials

Let $f\in \mathbb{Z}[x_1,\dots,x_n]$ be a polynomial. I want to show that a certain monomial $m$ shows up with non-zero coefficient in the $r^{th}$ power of $f$. If you're lucky, you can do this as ...
7
votes
1answer
824 views

Recalling result of tensor product of polynomial rings

Let $k$ be a field (alg closed if you want). Now let $I_{i}$ be an ideal of $k[x_{i}]$ for every $i \in \{1,2,\ldots,n\}$. Is it always true that: $$k[x_1,x_2,\ldots,x_n]/ \langle I_1,I_2,\ldots,I_n ...
1
vote
1answer
128 views

A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course ...
2
votes
1answer
113 views

Question on a transfinite construction in algebra

I am studying out of Matsumura's Commutative Ring Theory, and in the first section on modules he proves (following Kaplansky) that every projective module over a local ring is free. My questions have ...
3
votes
2answers
194 views

a ring of fractions which has finitely many maximal ideals

Let $R$ be a commutative ring and $P_1,\ldots ,P_n$ be prime ideals of $R$. If $S=\bigcap_{i=1}^n (R\setminus P_i)$ then show that the ring of fractions $S^{-1}R$ has only finitely many maximal ...
5
votes
1answer
319 views

Nilradical of a primary ideal is a minimal prime

I'd like to show the following claim: The radical of a primary ideal $\mathfrak q$, $r(\mathfrak q)$, is the smallest prime ideal containing $\mathfrak q$. Can you tell me if my proof is correct: ...
2
votes
1answer
308 views

Proving $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without the universal property

Let $F$ be a commutative field, and let $U$, $V$, and $W$ be finite dimensional vector spaces over $F$. How can one prove $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$ without using the ...
3
votes
1answer
109 views

Powers of primes in a Dedekind domain

I saw a number theory problem that began with "Fix $\pi \in Q - Q^2$; then $\pi^m \in Q^m - Q^{m+1}$." Here $Q$ is a prime ideal of a Dedekind domain, so this is obvious by unique prime factorization. ...
131
votes
0answers
4k views

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on Mathoverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
10
votes
2answers
883 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
3
votes
2answers
329 views

A question about Euclidean Domain

This is a problem from Aluffi's book, chapter V 2.17. "Let $R$ be a Euclidean Domain that is not a field. Prove that there exists a nonzero, nonunit element $c$ in $R$ such that $\forall a \in R$, ...
5
votes
2answers
256 views

Example of an application of a theorem about ideals in rings of fractions in Atiyah-MacDonald

In Atiyah-MacDonald, we have the following theorem (p. 41): Proposition 3.11. i) Every ideal in $S^{-1}R$ is an extended ideal. ii) If $I$ is an ideal in $R$ then $I^{ec} = \bigcup_{s \in S} (I : ...
1
vote
1answer
61 views

Alternative proof or typo?

In Atiyah-MacDonald, we have the claim that $S^{-1}(I+J) = S^{-1}I + S^{-1}J$ and similarly, $S^{-1}(IJ) = S^{-1}I S^{-1}J$. Here $I,J$ are ideals of a commutative unital ring $R$ and $S$ is a ...
1
vote
1answer
108 views

Is there a name for this ideal constructed in terms of two submodules?

If $M$ is an $R$-module and $M_1, M_2$ are submodules of $M$, then one can construct the ideal $\{ r \in R \mid rM_2 \subseteq M_1 \}$, which is denoted $(M_1 : M_2)$. Does this construction have a ...
6
votes
1answer
235 views

Necessary and sufficient condition that a localization of an integral domain is integrally closed

Is the following proposition true? Proposition? Let $A$ be an integral domain, $K$ its the field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated ...
4
votes
1answer
489 views

The contraction of a maximal ideal of $A[[x]]$ is a maximal ideal of $A$?

I am working on the problems in Atiyah and MacDonald's famous Introduction to Commutative Algebra. On p. 11, problem 5 part iv reads: [Show that] the contraction of a maximal ideal $\mathfrak{m}$ ...
4
votes
2answers
230 views

$k[x]\otimes_k k[x]\cong k[x,y]$?

I think I am finally beginning to understand tensor products of algebras, and I could use a reality check. If I am understanding correctly, then $k[x]\otimes_k k[x]$ is ring-isomorphic to $k[x,y]$ by ...
2
votes
2answers
189 views

Is the ring of integers in a relative algebraic number field faithfully flat over a ground ring?

Let $L$ be a finite extension of an algebraic number field $K$. Let $A$ and $B$ be the rings of integers in $K$ and $L$ respectively. Is $B$ faithfully flat over $A$? What if $L$ is an infinite ...
9
votes
1answer
194 views

Lifting isomorphisms between derived categories

Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. I will denote by $D(A)$ and $D(B)$ the derived categories of unbounded complexes over $A$ and $B$. Suppose ...
3
votes
1answer
256 views

Localization arguments in Dedekind domains

I am reading Serre's Local Fields, and have questions about the text. Specifically, pages 11 and 12. 1) Consider a Dedekind domain. We want to show that all fractional ideals are invertible. Serre ...
4
votes
1answer
120 views

Taking fractions $S^{-1}$ commutes with taking intersection

Let $N,P$ be submodules of an $R$-module $M$ and let $S$ be a multiplicative subset of $R$. I think I proved $S^{-1}(N \cap P) = S^{-1}N \cap S^{-1} P$ but since my proof is not the same as the one ...
5
votes
1answer
90 views

If $S$ consists of units then $S^{-1}R \cong R$

I want to show that if $S$ consists of units then $S^{-1}R \cong R$. Can you tell me if my proof is correct? Since $S$ consists of units, $S$ is zero-divisor free and hence $f: R \to S^{-1}R$, $r ...
4
votes
2answers
416 views

Characterization of ideals in rings of fractions

Let $R$ be a commutative unital ring. Let $S$ be a multiplicative subset. Is there a characterisation of the ideals in the ring of fractions $S^{-1}R$ in terms of ideals $I$ in $R$ and $R$?
2
votes
3answers
349 views

Examples of rings of fractions

I wanted to come up with a few examples of rings of fractions $S^{-1}R$. Can you tell me if these are correct: 1.Let $R = \mathbb Z$, $S = (2 \mathbb Z \setminus \{0\}) \cup \{1\}$. Then every $[x] = ...
3
votes
2answers
444 views

Tensor product of $R$-algebras

Let $f: R \to S$ and $g: R \to T$ be two $R$-algebras. To show that $S \otimes_R T$ is an $R$-algebra I need to define a ring structure (multiplication) on it and a ring homomorphism $h : R \to S ...
3
votes
3answers
365 views

$R \otimes_R M \cong M$

Let $R$ be a commutative unital ring and $M$ an $R$-module. I'm trying to prove $R \otimes_R M \cong M$ but I'm stuck. If $(R \otimes M, b)$ is the tensor product then I thought I could construct an ...
8
votes
2answers
244 views

Proposition 5.21 in Atiyah-MacDonald

There's just one step in this proof I can't see for the life of me. Set up: We have a field K and an algebraically closed field $\Omega$. $(B, g)$ is maximal in the set $\Sigma$ of ordered pairs ...
2
votes
1answer
134 views

Example computation of $\operatorname{Tor_i}{(M,N)}$

Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$. Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct: (i) First we want a projective resolution of ...
2
votes
1answer
179 views

Question about a proof of $f$ injective $\implies$ $f \otimes \operatorname{id}$ injective

I'd like to prove (i) implies (ii) where: (i) Whenever $f: A \to B$ is injective and $A,B$ are finitely generated then $f \otimes \operatorname{id}: A \otimes P \to B \otimes P$ is injective. (ii) ...
0
votes
1answer
315 views

Question about flat modules and exact sequences

I have a basic question about exact sequences. I want to show that if I have that whenever $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is exact then $0 \to A \otimes N \to B \otimes N \to C ...
2
votes
1answer
178 views

Existence of an element of given orders at finitely many prime ideals of a Dedekind domain

Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of $A$. Let $\alpha \in A$. Let $k$ be a non-negative integer. If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = ...