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

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2
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1answer
60 views

Help with Math software (macaulay 2)

I just started working with Macaulay 2 and need some help. I need to find the number of solutions of a system of equations. I am having difficulty imputing this into the software so please be specific ...
0
votes
1answer
18 views

Is an $R$-module $A$ a module over the image of a homomorphism $f:R\rightarrow{f(R)}?

Let $R$ and be a (unital) commutative ring with $A$ as an $R$-module. Now suppose $f\in{Hom(R,f(R))}$ I am wondering what requirements (if any exist) need to be placed on $f$ to ensure that $A$ is an ...
2
votes
0answers
31 views

Rings where action of automorphisms on maximal ideals is transitive

If $R$ is a commutative ring, $\alpha: R \to R$ an automorphism of $R$, and $M$ a maximal ideal of $R$, then $\alpha(M)$ is also a maximal ideal of $R$ with the same quotient field. So the group of ...
5
votes
3answers
77 views

is a number field by definition a subfield of $ \mathbb C $?

I have seen that some authors are defing the number field as a subfield of $ \mathbb C$ which is a finite extension of the rational numbers $ \mathbb Q $, while some others without referering to ...
0
votes
1answer
62 views

Localization commutes with direct sum.

Let $A$ be a commutative ring and $S \subseteq A$ a multiplicative subset. If $N$, $M$ are $A$-modules, is it true that $S^{-1} M \oplus S^{-1} N \simeq S^{-1} (M \bigoplus N)$? I need an ...
2
votes
2answers
81 views

Annihilators of elements of a finitely generated faithful module over a noetherian reduced ring

Lately I've been thinking to annihilator of modules and I've conjectured a proposition I can't prove, so I'll expose my claim. Let $A$ be a noetherian reduced (commutative) ring and let $M$ be a ...
1
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1answer
37 views

If $A$ is a Krull ring of $K$, then $A \cap K'$ is also Krull for $K' \subset K$.

If $A$ is a Krull ring of $K$, then $A \cap K'$ is also Krull for $K' \subset K$ (where $K'$ is a subfield of $K$). What's confusing me is that $K'$ may not contain the uniformising element $t$ ...
3
votes
2answers
73 views

Prerequisite of Algebraic Geometry

Algebraic geometry, as far as I know, is a very important branch of mathematics, which is also very difficult. I am going to take a try to taste that. Before really going into the field, I have two ...
3
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1answer
112 views

Reference request: (categorical) commutative algebra text

I'd like a text that puts commutative algebra in a categorical framework. I'm wondering if anybody has any recommendations.
-1
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1answer
53 views

Counterexamples for correspondence theorem of a localization $R\to S^{-1}R$ [closed]

Let $R$ be a commutative unitary ring an $S\subseteq R$ a multiplicatively closed subset. Consider the localization homomorphism $\varphi:R\to S^{-1}R$. There is a one-to-one correspondence between ...
0
votes
1answer
34 views

Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings?

Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings? Is it of finite type? Is the map $\operatorname{Spec} \prod_{i=1}^\infty \mathbb{C} ...
6
votes
1answer
64 views

Examples of rings whose polynomial rings have large dimension

If $A$ is a commutative ring with unity, then a fact proved in most commutative algebra textbooks is: $$\dim A + 1\leq\dim A[X] \leq 2\dim A + 1$$ Idea of proof: each prime of $A$ in a chain can ...
0
votes
2answers
44 views

Krull dimension of polynomial rings over noetherian rings

I want to prove the following theorem concerning Krull dimension: Theorem If $A$ is a noetherian ring then $$\dim(A[x_1,x_2, \dots , x_n]) = \dim(A) + n$$ where $\dim$ stands for the Krull ...
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3answers
49 views

Ideal of Ring of holomorphic functions

Can you tell me a non trivial ideal of ring of holomorphic functions from C to C.
2
votes
1answer
25 views

On general topological spaces and $C(X, \mathbb R)$ , where for closed sets $A,B$ in $X$ , $I_A=I_B \implies A=B$

Let $X$ be a metric space and $C(X, \mathbb R)$ be the ring of all real valued continuous functions from $X$ . For $A \subseteq X$ , let us define $I_A :=\{f \in C(X, \mathbb R) : f(x)=0 , \forall x ...
0
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0answers
19 views

factor out of an expression, a couple principal ideals, software?

I have an expression, $f$, consisting of a few rational fractions of large multivariate numerators, $n1,\,n2,\ldots \in \mathbb{Q}[a1,a2,b1,b2;Q]$ and large multivariate denominators, $d1,\,d2,\ldots ...
1
vote
1answer
22 views

Castelnuovo-Mumford regularity of a shifted module

Let $R$ be a graded ring and $M$ be a graded $R$-module. Regularity definition $\operatorname{reg}(M)=\max\{j-i\mid \beta _{i,j}(M) \not=0\} $. What is the relation between ...
1
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1answer
41 views

A radical ideal in a commutative ring is prime if and only if it is not an intersection of two radical ideals properly containing it?

Let $I$ be a radical ideal (i.e. $\sqrt I=I$) in a commutative ring with unity. Then is it true that $I$ is a prime ideal if and only if it is not an intersection of two radical ideals properly ...
1
vote
1answer
26 views

Does the nilpotent extension of a $1$-dimensional algebra always give a projective module?

Let $A$ be a $1$-dimensional reduced Noetherian algebra over an algebraic closed field $k$ with characteristic zero. Let $(B,N)$ be a nilpotent extension of $A$, i.e. $B$ is a Noetherian $k$-algebra, ...
3
votes
1answer
34 views

Depth comparison 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
37 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 ...
5
votes
1answer
107 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. ...
2
votes
2answers
31 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 ...
2
votes
1answer
96 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 $M$ ...
3
votes
1answer
141 views

Proving exactness of the conormal sequence

Problem: 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} ...
3
votes
1answer
32 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 ...
2
votes
1answer
17 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 $f(t)/\prod_{i=1}^{s}(1-t^{k_{i}})$ ...
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2answers
61 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
101 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
votes
3answers
110 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
40 views

Power series with coefficients in primary ideals

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
23 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 ...
2
votes
0answers
29 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
57 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
votes
1answer
31 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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vote
2answers
57 views

Local ring inside a function field of transcendence degree one

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 $R$ ...
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votes
0answers
52 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
45 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
votes
1answer
60 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
56 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 ...
3
votes
2answers
42 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
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1answer
22 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
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1answer
43 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
48 views

A problem on the associated primes of a 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 $R$. ...
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1answer
77 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
71 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
59 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
68 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
59 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
29 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 ...