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

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0
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2answers
81 views

Proving $\operatorname{Spec} k[x_1,\cdots, x_r] \subset \operatorname{Spec} k[x_1,\cdots, x_{r+1}]$

Let $k$ be a field. Is there an elegant proof of the fact that $\operatorname{Spec} k[x_1,\cdots, x_r] \subset \operatorname{Spec} k[x_1,\cdots, x_{r+1}]$? I proved it as follows: let $P \in ...
0
votes
1answer
103 views

A zero-dimensional ring is Noetherian?

Proposition: Let $A$ be a non-zero ring that is not a field. Suppose $A$ is zero dimensional. Then it is Noetherian. Proof: Let $p$ be a prime ideal of $A$. If $p$ is not maximal, then $p \subsetneq ...
1
vote
1answer
251 views

Prove that if $M$ is an $R$-module of finite length, then $\operatorname{End}_R(M)$ is artinian

Prove that if $M$ is an $R$-module of finite length, then $\operatorname{End}_R(M)$ is artinian. We can derive that $M$ is both artinian and noetherian from that it has finite length, and its ...
3
votes
1answer
143 views

Integral extension (Exercise 4.9, M. Reid, Undergraduate Commutative Algebra)

Let $k$ be any field and let $A = k[X,Y,Z]/(X^2 - Y^3 - 1, XZ - 1)$. How can I find $\alpha, \beta \in k$ such that $A$ is integral over $B = k[X + \alpha Y + \beta Z]$? For these values of ...
2
votes
2answers
271 views

Modules over local ring and completion

I'm stuck again at a commutative algebra question. Would love some help with this completion business... We have a local ring $R$ and $M$ is a $R$-module with unique assassin/associated prime the ...
2
votes
1answer
935 views

Finitely generated modules over artinian rings have finite length

Suppose $M$ is an $R$-module. Prove that $M$ has finite length if $R$ is artinian and $M$ is finitely generated.
7
votes
3answers
310 views

When does locally irreducible imply irreducible?

The situation is this: I have a homogeneous ideal with many generators and variables, too many to simply ask isPrime I in Macaulay2. However, the ideal simplifies ...
6
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2answers
549 views

Ring of formal power series finitely generated as algebra?

I'm asked if the ring of formal power series is finitely generated as a $K$-algebra. Intuition says no, but I don't know where to start. Any hint or suggestion?
2
votes
1answer
51 views

Question on Integral Closure

I'm trying to prove this fact: given $A$ an integral domain and an element $f\in A$ such that $A/fA$ has no nilpotents, then $A$ is integrally closed if and only if $A_f$ is integrally closed ...
2
votes
1answer
110 views

Valuation over the algebraically closed field of rational number

How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?
0
votes
1answer
45 views

If ideals $Q_1,Q_2$ lie over a prime in $\Bbb{Z}$ their product lies over the prime squared?

Suppose we have a Dedekind domain $R$ which for the moment we can take to be $\mathcal{O}_K$ for some algebraic number field $K$. Now suppose that $Q_1,Q_2$ are prime ideals that lie over a prime ...
1
vote
2answers
117 views

Adjoining an inverse to a local UFD of Krull dimension 2 gives a PID

Let $R$ be a local UFD of Krull dimension 2. Let $a\in R$ be a nonzero, non-unit. I am trying to show that the ring $R[1/a]$ is a principal ideal domain. Does anyone have any suggestions as to how ...
4
votes
1answer
192 views

Is a surjective homomorphism of regular local rings necessarily an isomorphism?

Let $R$ and $S$ be regular local rings, and $f: R\rightarrow S$ a surjection that induces an isomorphism on tangent spaces. Is $f$ necessarily an isomorphism? I believe the answer should be yes, ...
3
votes
2answers
150 views

When can a ring imbed in a localization?

In Topics in Algebra, there is an exercise (3.6.5): let $R$ be a commutative, unital ring and let $S\subset R$ be non-empty and such that $s_1 s_2\in S$ if $s_1,s_2\in S$ and $0\not\in S$. Construct ...
1
vote
1answer
308 views

$(M\otimes_A N)_B \cong M_B\otimes_B N_B$?

Let $A \rightarrow B$ be a homomorphism of commutative rings. Let $M, N$ be $A$-modules. We denote $M\otimes_A B$ by $M_B$. We regard $M_B$ as a $B$-module. Then $(M\otimes_A N)_B \cong M_B\otimes_B ...
1
vote
2answers
91 views

a commutative ring statisfying an integer polynomial

Let $p(X)\in\mathbb{Z}[X]$ be a monic polynomial and let $A$ be a commutative ring in which every element is a zero of $p(X)$. Prove that all prime ideals in $A$ are maximal. By definition, give ...
6
votes
5answers
556 views

Is there a finitely generated, algebraic $K$-algebra $A$ that is not a field?

There is a well-known theorem that states that if $A$ is a finitely generated $K$-algebra, an integral domain and algebraic over $K$, then $A$ is a field. Is the integral domain condition necesary? I ...
5
votes
5answers
185 views

A question on local rings

I was trying to get a counterexample of this fact: given a ring $A$, $f\in A$ and $S=\{1,f,f^2,...\}$, is $S^{-1}A$ always a local ring? Could you help me please? Thank you.
2
votes
2answers
188 views

Irreducibility and Subspace Topology

Is the following statement true? "Let $X$ be a topological space, $Y$ a subspace and $S$ a closed and irreducible subset of $X$. Then $Y \cap S$ is not necessarily irreducible in $Y$." Counterexamples ...
2
votes
2answers
86 views

Can a ring of integers contain a $2$-dimensional noetherian normal integral domain?

Let $K$ be a number field with ring of integers $O_K$. Does there exist a $2$-dimensional subring $A\subset O_K$? Clearly, if such a subring $A\subset O_K$ exists, we have that $A$ is an integral ...
6
votes
2answers
1k views

In a finite ring extension there are only finitely many prime ideals lying over a given prime ideal [duplicate]

I'm trying to solve the exercise 6.7 of Miles Reid's Undergraduate Commutative Algebra (pag 93). How can I prove that if $B$ is a finite ring extension of $A$, there are only finitely many prime ...
6
votes
1answer
171 views

Vanishing of local cohomology $\operatorname{H}^1_J(\Gamma_I(M))=0$

Let $M$ be a module over Noetherian ring $R$ such that $\operatorname{H}^1_I(M)=0$ for every ideal $I$ of $R$. Show that $\operatorname{H}^1_J(\Gamma_I(M))=0$ for every ideal $J$. I tried to prove it ...
9
votes
1answer
291 views

Isomorphism of rings implies isomorphism of vector spaces?

Let $A$ and $B$ be isomorphic unitary rings. Suppose that both of them admit a structure of (maybe finite dimensional) vector space over some field $k$. I would like to know if then $A$ and $B$ are ...
6
votes
1answer
240 views

Bounded index of nilpotency

A ring $R$ is said to have a bounded index (of nilpotency) if there is a positive integer $m$ such that $x^m = 0$ for every nilpotent $x\in R$. I wonder whether this property transfers to the ring of ...
1
vote
1answer
82 views

Cohomological dimension and height of ideals

A well-known fact: Let $R$ be a Noetherian ring and $I$ an ideal. Then $${\rm ht}(I) \leq {\rm cd}(I,R).$$ I looked, but could not find the proof of this fact. I want to see the proof of ...
3
votes
1answer
580 views

Support of a module and submodule

$\newcommand{\Supp}{\operatorname{Supp}}$ $\newcommand{\Spec}{\operatorname{Spec}}$ $\newcommand{\Ann}{\operatorname{Ann}}$ Let $A$ be a commutative ring with 1. Let $M$ be a finitely-generated ...
19
votes
3answers
381 views

Bound on nilpotency index of endomorphisms

Let $A$ be a Noetherian ring (commutative with $1$) and $M$ a finitely generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism $T : M \to M$ we ...
5
votes
3answers
483 views

Expressing, in terms of $I$ and $M$, the $R$-modules $\mathrm{Hom}_R(R/I,M)$, $\mathrm{Hom}_R(M,R/I)$, $\mathrm{Hom}_R(I,M)$, $\mathrm{Hom}_R(M,I)$

Let $R$ be a commutative unital ring, $I$ an ideal of $R$, and $M$ a $R$-module. It is known that $R/I \otimes_R M \cong M/IM$. Also, $\mathrm{Hom}_R(R,M)\cong M$. Is there some similar formula for ...
8
votes
1answer
333 views

Ideals generated by regular sequences and the vanishing of $\operatorname{Tor}$

Let $(R,\mathfrak m)$ be a Noetherian local ring and $I$, $J$ two ideals of $R$ such that $I$ is generated by an $R/J$-sequence (this means $I=(x_1,\dots,x_t)$ where $x_1,\dots,x_t$ is an ...
4
votes
0answers
128 views

Hilbert symbol over a ring

Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows: $$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
5
votes
1answer
206 views

Examples of extensions of a perfect field which are not separably generated

Let $K$ be an extension field of a field $k$. We say $K$ is separably generated over $k$ if $K$ has a transcendence basis $S$ over $k$ such that $K$ is separably algebraic over $k(S)$. Let $k$ be a ...
4
votes
1answer
161 views

Axioms for the Exterior Derivative

On Serge Lang's book Algebra, he wrote on page 748, after defining the module of differentials $\Omega^1_{A/R}$ of an $R$-algebra $A$ and higher differentials $\Omega^i_{A/R}=\wedge^i \Omega^1_{A/R}$, ...
7
votes
1answer
300 views

Localization of a UFD that is a PID

Let $R$ be a UFD and $r\in R$ irreducible. If $S=R-(r)$, why is $S^{-1}R$ a PID? I've finished proving the integral domain part, which was pretty easy. How to prove it's principal? consider $R ...
2
votes
1answer
183 views

Why if the maximal ideal is nilpotent then this module is free?

On matsumura is proved the following proposition: let $(R,m)$ be a local ring and $M$ a flat $R$-module. If $x_1,\ldots,x_n\in M$ are such that their images $\bar{x}_1,\ldots,\bar{x}_n$ in ...
1
vote
1answer
145 views

Valuation rings of complete non-archimedean fields which are not local

I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
2
votes
2answers
91 views

Is every subalgebra of $F[x]$ one-dimensional? [duplicate]

Let $F$ be a field. Let $A$ be a subalgebra of the polynomial ring $F[x]$. Does $A$ necessarily have Krull dimension $\leq 1$?
3
votes
1answer
123 views

tensoring a polynomial algebra: $R[x_i]/\mathfrak{a}\otimes A \cong A[x_i]/\mathfrak{a}$

Let $R$ be a commutative unital ring, $A$ an associative unital $R$-algebra, $I$ an arbitrary set, and $\mathfrak{a}$ an ideal of $R[x_i; i\!\in\!I]$. If $A$ is commutative, then there is an ...
2
votes
1answer
91 views

Geometrically reduced algebraic extension of a field

Let $k$ be a field. Let $A$ be a commutative algebra over $k$. We say $A$ is geometrically reduced over $k$ if $A\otimes_k k'$ is reduced for every extension $k'$ of $k$. Let $K$ be an algebraic ...
0
votes
1answer
192 views

Problem on two ideals and their quotient

For two ideals $I$ and $J$ in a commutative ring $R$, define $I : J = \{a\in R : aJ \subset I\}$. In the ring $\mathbb{Z}$ of all integers, if $I = 12\mathbb {Z}$ and $J = 8\mathbb {Z}$, find $I : ...
4
votes
2answers
421 views

Tangent spaces of affine algebraic varieties at singular points

Let $X$ be an affine algebraic over the algebraically closed field $k$ and let $\mathcal{O}(X)$ be the ring of its regular functions. Let us assume that $X$ is irreducible and let $x\in X$. There ...
3
votes
2answers
379 views

Hilbert-Samuel polynomial computation tutorial

I'm trying to understand how to compute the Hilbert-Samuel polynomial of a specific example. Could someone help me with an elaborate computation so that I get it... For example, what is the ...
3
votes
2answers
154 views

Localization of ring by using homomorphism

If $\phi : A \to B$ is a ring homomorphism, where $A$ and $B$ are commutative rings with unit but not necessarily domains. Let $P$ be a prime ideal of $A$. How do we define the ring $B_{P}$ and map ...
7
votes
1answer
340 views

The support of a non finitely generated module

Let $R$ be a ring and $M$ an $R$-module. If we suppose $M$ finitely generated then (following Matsumura) if we write $M=Rm_1+\cdots+Rm_n$ we have: $p\in\mathrm{Supp}\;M$ if and only if $M_p\neq0$ if ...
1
vote
1answer
114 views

A question on localization

For an integral domain $R$, I know that $$R=\bigcap_{\text{maximal ideals }\mathcal{m}}R_{\mathcal{m}}.$$ Why must $R$ be an integral domain? I want to know a counterexample when $R$ is not an ...
0
votes
0answers
84 views

Primary extensions of a field

An extension $L$ of a field $k$ is called primary if the relative algebraic closure of $k$ in $L$ is purely inseparable over $k$. I'm looking for a proof of the following proposition which is used in ...
3
votes
1answer
70 views

filtered colimit of $Hom_{A_i}(M_0\otimes_{A_0} A_i, N_0\otimes_{A_0} A_i)$

Let $I$ be a small filtered category. Let $F\colon I \rightarrow \textbf{CRng}$ be a functor, where $\textbf{CRng}$ is the category of commutative rings. We write $A_i = F(i)$ for $i \in I$, $A =$ ...
4
votes
1answer
224 views

Finite field extensions of $\mathbb C(t)$ are quotient fields of $\mathbb C[X,Y]$ modulo an irreducible polynomial?

Given a field of the form $\mathbb C(t)[g]$, where $t$ is transcendental over $\mathbb C$ and $g$ is algebraic over $\mathbb C(t)$ do I always find an irreducible polynomial $F\in \mathbb C[X,Y]$ such ...
0
votes
1answer
85 views

Nonvanishing of Hom and tensor product

Let $R$ be a commutative ring and $M\neq 0$ an $R$-module. Show that if for every $R$-module $X\neq 0$, $\operatorname{Hom}_R(M,X)\neq 0$ then $M\otimes_R X\neq0$ for every $R$-module $X\neq 0$.
0
votes
1answer
105 views

Determinant of matrices with independent indeterminates entries

Let $R$ be a unique factorization domain, let $X=(x_{ij})_{1\leq i\leq n,1\leq j\leq n}$, be a family of independent indeterminates, and let $R_{nn}$ be the polynomial ring ...
1
vote
1answer
98 views

On colim $Hom_{A-alg}(B, C_i)$

We assume all rings considered are commutative. Let $A$ be a ring. Let $B$ be an $A$-algebra of finite presentation. Let $I$ be a small filtered category. Let $C\colon I \rightarrow$ $A$-alg be a ...