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

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2
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1answer
17 views

Depth comparision on short exact sequences

Let $R$ be a Noetherian ring and $M,N,U$ be $R$-modules. We have a short exact sequence $$0 \longrightarrow U \longrightarrow M \longrightarrow N \longrightarrow 0.$$ We know that ...
2
votes
2answers
23 views

Is $\mathbb Z[\frac{1+\sqrt 5i}2]$ a ring of fractions of $\mathbb Z[\sqrt 5i]$?

Does there exist a multiplicative set $S\subset \mathbb Z[\sqrt 5i]$ such that $\mathbb Z[\frac{1+\sqrt 5i}2]\cong S^{-1}\mathbb Z[\sqrt 5i]$? Since the multiplicative structure of $\mathbb ...
3
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0answers
34 views

Stable epimorphisms of commutative rings

Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism. ...
1
vote
2answers
17 views

Find $\operatorname{depth}(R/p_1\cap p_2)$

Let $R=K[x_1,\dots,x_n]$ be a polynomial ring over a field, $K$. Let $I$ be a square free monomial ideal of $R$. Let $p_1 ,p_2$ be minimal prime ideals of $I$ generated by subsets of ...
1
vote
1answer
33 views

An exercise using Nakayama's lemma.

Let $A$ be a ring and $\mathfrak a \subseteq A$ an ideal. Let $N \to M$ be a homomorphism of $A$-modules such that the induced homomorphism $N/\mathfrak a N \to M/\mathfrak a M$ is surjective. If $M$ ...
2
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0answers
25 views

Proving exactness of the conormal sequence

Let $\phi \colon A \to B$ be a surjective homomorphism of $R$-algebras with kernel $I$. I want to show that the conormal sequence $$ I/I{}^2 \longrightarrow B \otimes_A \Omega_{A/R} \longrightarrow ...
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0answers
44 views

Is $k[x,y]=k\oplus \langle x,y\rangle$? [on hold]

Is $k[x,y]=k\oplus \langle x,y\rangle$? If it's not right, could you give me an counter example? Thank you!
2
votes
1answer
23 views

$(A_f)_{g/f^{n_0}}\cong A_{fg}$ (localization with the powers of an element)

I'm working in a problem from Hartshorne Algebraic Geometry. But I need a result from Commutative Algebra. Given a commutative ring $B$ with $1$. For each $b \in B$ define the ring $B_b$ as the ...
1
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1answer
16 views

A question on additive-functions in the proof of the Hilbert-Serre Theorem

I am trying to understand the proof of the following Theorem from Atiyah-MacDonald. $P(M,t)$ is a rational function in t of the form $f(t)/\prod_{i=1}^{s}(1-t^{k_{i}})$ ...
1
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2answers
29 views

An exercise on tensor product over an integral domain.

This post is the natural conclusion of another one (An exercise on tensor product over a local integral domain.). Let $M$ be a finite module over an integral domain $A$. Let $Q$ be its fraction ...
3
votes
1answer
79 views

Is it true that $R^n\simeq R^m$ as rings implies $m=n$?

Let $R$ be a commutative ring. We know that if $R^n\simeq R^m$ as $R$-modules for some positive integers $n,m$ then $n=m$. But is it still true when they are isomorphic as rings? Thanks!
3
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3answers
87 views

Finite commutative ring with more than $\frac{2}{3}$ of its elements idempotent

Suppose that $R$ is a finite commutative ring with identity element, such that more than $\frac{2}{3}$ of elements are idempotent. Prove that all of elements are idempotent. Please give me a ...
1
vote
1answer
30 views

Power series with coefficients in primary rings

Let $P$ be a prime ideal in a commutative ring $R$ with unity such that an ideal $Q$ is $P$-primary and some power of $P$ is a subset of $Q$. I want to show that $\sqrt {Q[[x]]}=P[[x]]$. If a ...
1
vote
1answer
18 views

What is the meaning of 'homogeneous' here? And what does it mean by 'degree'?

This is a part extracted from a textbook that has many definitions that I was confused and failed to find. Let $\displaystyle A=\oplus_{n=0}^\infty A_n$ be a Noetherian graded ring. Then $A_0$ is ...
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0answers
25 views

Proof verification: $Hom_A(P,M) \cong Hom_A(P,A)\otimes_A M$

Let $A$ be a commutative noetherian ring, let $P$ be a free $A$-module, and let $M$ be a finitely generated $A$-module. I want to show that $Hom_A(P,M) \cong Hom_A(P,A) \otimes_A M$. Here is my ...
3
votes
1answer
46 views

Injection and surjection over free modules.

Let $A$ be a commutative ring and $M$ an $A$-module. Suppose to have both an injection $A^s \to M$ and a surjection $A^s \to M$ of module homomorphisms. Show that $M \simeq A^s$. This point is ...
0
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1answer
28 views

Question on Quotient Rings and moding

Hi guys just a quick question If we have an integral domain and we mod it with an irreducible element such as $K[x,y,z]/<p(x,y,z)>$ where p is irreducible polynomial then the resulting ...
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2answers
35 views

Local ring inside a function field of trans deg 1

Let $K$ be a function field of transcendence degree 1 over a base field $k$. Let $(R,\mathfrak{m}) \subseteq K$ be a local ring that is not a field. Suppose $S,T$ are DVR's of $K$ which dominate $R$ ...
3
votes
0answers
29 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
2
votes
1answer
42 views

dimension of quotient by algebraically independent elements

Let $f_1,\dots,f_s$ be algebraically independent polynomials of $A:=k[x_1,\dots,x_n]$, $s \le n$. Recall that algebraically independent means that there is no non-zero polynomial $g \in ...
3
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1answer
52 views

Algebraic independence via the Jacobian

I have seen being mentioned that algebraic independence of polynomials can be tested by the so called Jacobian Criterion (Apparently one takes the Jacobian matrix of these polynomials and inspects the ...
0
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1answer
31 views

Extending regular function on normal variety from a subvariety of codimension 2

In his book "Commutative Algebra with a View Toward Algebraic Geometry" Eisenbud proves the Corollary 11.4 which states the following If $R$ is a normal Noetherian domain, then $R$ is the ...
2
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2answers
34 views

Linearly independent elements are less than generators in a module.

Let $R$ be a commutative ring and $M$ a finitely generated $R$-module. Let $s$ the maximum number of linearly independent elements of $M$, while $t$ is the minimum number of a system of ...
1
vote
1answer
19 views

zero element in tensor product of a localization ring and a module

Let $R$ be a commutative ring with $1$. Let $f$ be a non-nilpotent element of $R$ and let $R_f$ be a localization of $R$ by the multiplicative set $\{ f^i \mid i=0,1,2,\dots\}$. Let $M$ be an ...
0
votes
1answer
41 views

dimension inequality for graded versus non-graded polynomial rings

Let $A=k[x_1,\dots,x_n]$ be a polynomial ring over an algebraically closed field $k$. Let $I$ be an ideal of $A$ and $f$ some element of $A$. Then the Krull dimension does not necessarily satisfy the ...
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0answers
30 views

a problem on local cohomology module

The question comes from Proposition 2.1 in the paper "Associated primes of local cohomology module and Matlis duality". Let $R$ be a Noetherian local ring of dimension $d$ and $I$ an ideal of $R$. ...
1
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1answer
68 views

Injectivity Unclear

Let $R=K[x_1,...,x_n]/I$ and $m$ be maximal ideal of $R.$ Let $(s_1,...,s_d)$ be a base of $m/m^2$ where $\dim R_m=\dim_K m/m^2=d.$ Then by Kunz Chapter V.5.10 the canonical epimorphism ...
1
vote
1answer
47 views

An exercise on tensor product over a local integral domain.

Let $M$ be a finite module over a local integral domain $(A,m)$. Let $k$ be its residue field and $Q$ its fraction field. Consider the $k$-vector space $M \otimes_A k$ and the $Q$-vector space $M ...
2
votes
2answers
52 views

What is an example of a homomorphism of rings that doesn't preserve gcd's?

Given a commutative ring $R$, we say that $x$ is a gcd of $(y,z)$ iff the following conditions hold: $x \mid y,z$ For all $x' \in R$, if $x' \mid y,z$, then $x' \mid x$. This gives a ternary ...
3
votes
2answers
58 views

Commuting of Hom and Tensor Product functors?

Let $V_i,W_i$ be finite dimensional vector spaces, for $i=1,2$. Assume we have homomorphisms $\phi_i:V_i\rightarrow W_i$. Then, there is an induced map $\widehat{\phi_1 \times \phi_2} \in Hom(V_1 ...
2
votes
1answer
50 views

Dickson's Lemma

I am doing a course in Commutative algebra and there is a lemma called Dickson's lemma which states the following: Let $\mathfrak{I} = \langle X^{u}: u \in A\rangle$ for some set $A \subset ...
0
votes
1answer
19 views

Why is the degree condition for a degree reverse lexicographic order necessary?

A degree reverse lexicographic order $\prec$ is defined as follows: Given the polynomial ring $R=K[x_1,...,x_n]$. Two monomials in $R$ have the order $x^u\prec x^v$, if $\deg(x^u)<\deg(x^v)$, or ...
1
vote
1answer
31 views

Subsheaf of a torsion-free sheaf

Let $X$ be a noetherian projective scheme, $\mathcal{F}$ a torsion free $\mathcal{O}_X$-module on $X$ and $\mathcal{G} \subset \mathcal{F}$ submodule. Is it possible that $\mathcal{G}$ is ...
0
votes
3answers
44 views

Isomorphism between affine varieties

I am working with a ring and I am trying to show it is not isomorphic (as $k$-algebra) to another ring: $k[x,y,z]/\langle xy-z^2\rangle$ and $k[u,w]$. What I tried so far was. I aim for a ...
0
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3answers
74 views

Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
0
votes
1answer
34 views

Finding a bijective morphism

I am given two Varieties $Z=V(x^2+y^2+1) \subset C^2$ and $W=V(x^2-y^2-1) \subset C^2$. We need to find a bijective morphism f such is an isomorphism with the inverse of f. First how we defined ...
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0answers
29 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
2
votes
1answer
52 views

Isomorphism of Localizations

I believe, though a not sure, that any two ideals $A, B$ of a Dedekind domain $X$ are isomorphic as $X$-modules iff their localizations $A_p, B_p$ are isomorphic for any prime ideal $p$. Could anyone ...
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0answers
19 views

Not connected Zariski topology implies the existence of an idempotent element. [duplicate]

I am trying to prove that for a commutative ring $A$ (with the unit) the Zariski topology is not connected if and only if there is an idempotent element different from 0 and 1. I proved "Existence of ...
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1answer
24 views

Commutative Algebra and Monomial orders

So whenever we are doing any problem related to ideals in the polynomial ring $k[x_{1},x_{2},\dots x_{n}]$,(e.g. calculating a grobner basis for instance or doing the division algorithm for a set of ...
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0answers
21 views

Homogeneous prime ideal in $K[x_1,\ldots,x_n]$

Let $K$ be a field and $P$ a homogeneous prime ideal in $K[x_1,\ldots,x_n]$, with height $r$. I want to show that there is a chain of homogeneous primes $P_0\subsetneqq P_1\subsetneqq\cdots\subsetneqq ...
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1answer
22 views

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite.

Let $F$ be a field and $R$ a finitely generated $F$-algebra. Let $P$ be a maximal ideal of $R$. Then $\dim(R/P)$ as a vector space over $F$ is finite. $P$ is a maximal ideal of $R/P$ is a field. I ...
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0answers
46 views

Class number of $\mathbb Q(\sqrt{10}) $

I am interested in knowing how to compute the class number of $\mathbb Q(\sqrt{10}) $. I am confused with these class number computations.
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0answers
38 views

Krull's height theorem in the non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$. When $R$ is not Noetherian, this is not ...
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2answers
31 views

Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
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1answer
24 views

MaxSpec of the ring of continuous function on a compact topological space. [closed]

Consider a compact topological space $X$ and let $A$ be the ring of continuous functions $f: X \to \mathbb{C}$. Let $\mathfrak m_x$ for a point $x \in X$ be the kernel of the evaluation map $f \mapsto ...
2
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0answers
27 views

Ramification group - do you know/can produce a simple proof to this?

Let $(K,v)$ be a valuation field, $L$ a finite extension of $K$, and $w$ a valuation of $L$ above $v$. The ramification group of $w$ in $L$ is the subgroup of ${\rm Gal}(L/K)$ of all automorphisms ...
3
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1answer
28 views

Understanding the $\mathfrak{a}$-adic completion of an $A$-module as a functor

$\require{AMScd}$ I recently read the chapter 10 on Completions in Atiyah-MacDonald. They describe the $\mathfrak{a}$-adic completion $\hat{M}$ of an $A$-module $M$ as the inverse limit of an inverse ...
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2answers
84 views

On the minimal set of generators of monomial ideals in $\mathbb{C}[x,y]$.

I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry": i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of ...
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0answers
65 views
+50

Submodules $H$ satisfying: “if $ax \in H$ for some non-zero scalar $a$, then $x \in H$.”

Suppose $R$ is a commutative ring and that $X$ is an $R$-module. Question. Is there a term for those $R$-submodules $H$ of $X$ satisfying the following? For all $x \in X$, if $ax \in H$ ...