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

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8
votes
1answer
245 views

Separability and tensor product of fields

Is it true that a finite degree field extension $L/k$ is separable if and only if $L\otimes_{k}L$ is a reduced $L$-algebra? Surely the "only if" part is true because if the extension is ...
5
votes
1answer
193 views

A problem about localization of $\mathbb{Z}/6\mathbb{Z}$ at prime ideal $2\mathbb{Z}/6\mathbb{Z}$

We know that Given a prime ideal $P$ of a commutative ring, there is a one-to-one correspondence between $\lbrace\text{prime ideals }Q\subset P\rbrace$ and $\lbrace\text{prime ideals of } S^{-1}R ...
4
votes
2answers
77 views

does a prime in an extensions of integral domains remain radical?

Let $R\subset R'$ be an extension of integral domains. So we have an inclusion map $i:R\hookrightarrow R'$. Let $\mathfrak{p}\subset R$ be a prime ideal. We know that $\mathfrak{p}^e$ (generated by ...
1
vote
2answers
85 views

Question on the dimension $\dim 0$.

What is the Krull dimension $\dim 0$ of the trivial ring? Trivial ring is denoted by $0$
2
votes
2answers
122 views

$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
1
vote
1answer
38 views

example for $(\mathfrak{a}:\mathfrak{b})^e\subsetneq (\mathfrak{a}^e:\mathfrak{b}^e)$?

showing that the inclusion holds is an easy exercise, but can someone give an example where the inclusion $(\mathfrak{a}:\mathfrak{b})^e\subseteq (\mathfrak{a}^e:\mathfrak{b}^e)$ is strict?
0
votes
1answer
59 views

About direct sum and direct product of integral algebras

Is an easy exercise to prove the following assert: Let $B_0, \dots , B_n$ integral algebras over $A$. Then $\bigoplus_{i=0}^n B_i$ is integral over $A$. Is this true if the sum is infinite? What ...
3
votes
1answer
213 views

Projections are finite morphisms

Let $X$ be a variety in $\Bbb{P}^n$. I would like to see as simply as possible why the projection of $X$ from a point is a finite map. Suppose $p=(1:0:\ldots:0)\notin X$ and let ...
2
votes
3answers
276 views

Example of a commutative ring that is Artinian but not Noetherian

I want to give an example of a commutative ring that is Artinian but not Noetherian. Is there any examples not very difficult? I considered the ring $\mathbb{Z}+x\mathbb{Q}[x]$. It is not ...
1
vote
1answer
125 views

counterexample that $M$ is not finitely generated $R$-module and $M$ has no maximal submodule

In the proof of Nakayama Lemma, the following proposition is used: let $R$ be a commutative ring with identity and $M$ be a non-zero finitely generated $R$-module. Then $M$ has a maximal submodule. ...
7
votes
2answers
348 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
2
votes
1answer
89 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
7
votes
3answers
282 views

Does free functor preserve monomorphism?

The free functor is left adjoint to the forgetful functor so it preserves epimorphism. In the category of modules and algebras, it also preserves monomorphisms (the free functors being free modules ...
3
votes
1answer
91 views

Theorem 20.17 in Eisenbud: a few questions on the proof

Let $k$ be a field, $S=k[x_1,\dots,x_r]$ and $M$ a finitely-generated, graded $S$-module. Definition: We say that $M$ is weakly $m$-regular if $Ext^j(M,S)_n=0$ for all $j$ and $n=-m-j-1$. We say ...
8
votes
1answer
74 views

Disjoint standard open sets in Spec(R)

The following appeared as a homework problem last semester in Johan de Jong's algebraic geometry course at Columbia (http://www.math.columbia.edu/~dejong/schemes.html), described as "a bit of a ...
0
votes
1answer
133 views

Semilocal Rings and Noetherianity [duplicate]

How to prove that a semilocal ring such that all of its localizations at any maximal ideal are Noetherian is a Noetherian ring? For example, for the easier example, ...
1
vote
1answer
101 views

Being Galois stable under completion?

Let $R$ a Dedekind Domain, $K = \mathrm{Frac}(R)$ the fraction field, $L/K$ a finite galois extension, $R'$ the integral closure of $R$ in $L$. Then $R'$ is Dedekind again. Let $\mathfrak{p} \subset ...
4
votes
1answer
172 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
0
votes
1answer
62 views

On scalar extension of module and annihilator

Let $A, B$ be commutative rings with identity, $f: A \longrightarrow B$ a ring morphism, $M$ an $A$-module. Given $b\in B, x\in M$, does the following statement hold? $b\otimes x=0$ in $B ...
1
vote
2answers
296 views

Prime ideals in tensor products of algebras and their pullbacks

Suppose $\mathfrak{p}$ is a prime ideal in $B\otimes_CA$, and $\mathfrak{p}_A,\mathfrak{p}_B,\mathfrak{p}_C$ are its pullbacks in $A,B,C$. Does it hold: $(B\otimes_CA)_{\mathfrak{p}}\cong ...
2
votes
1answer
61 views

Nonexistence of a vector space isomorphism

I feel that the $\mathbf{Q}$ vector spaces $\prod_{n=0}^\infty \mathbf{Q}$ and $(\mathbf{Z}-0)^{-1}\prod_{n=0}^\infty\mathbf{Z}$ are not isomorphic, what is the quickest way to demonstrate it? By a ...
4
votes
1answer
252 views

Krull dimension of tensor product

Let $f: (R,m) \rightarrow (S,n)$ be a morphism of local Noetherian rings. Let $M$ be a finite $R$-module and $N$ a finite $S$-module such that $\operatorname{Supp}M = \operatorname{Spec} R$ and ...
5
votes
1answer
92 views

Maximal ideal not containg the set of powers of an element is prime

In the midst of attempting to prove that for a commutative ring $A$ with identity, and an ideal $I$ of $A$, $I = rad(I)$, where $rad(I) = \{x: x^m \epsilon I, m >0\}$, implies that $I$ is an ...
1
vote
1answer
134 views

Infinite direct product of rings free.

Let $A$ be a commutative ring (viewed as an $A$-module over itself) that is not a field. Are there some conditions that guarantee that $\prod_{k=0}^\infty A$ is free? What if $A=\mathbf{Z}$ or more ...
1
vote
2answers
269 views

Questions about a commutative ring with exactly three ideals

Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$ 1) Show that if $a \in R-I$, then $a$ is a unit in $R$. 2) Let $a,b\ne0$ in ...
1
vote
0answers
36 views

$M\cong N$ iff $[M:N]_R$ is a principal fractional ideal

Let $R$ be a Dedekind ring, $K$ its field of fractions, $U$ a finite vector space over $K$, and $M,N$ finitely generated $R$-modules that span $U$, i.e. contain a basis of $U$. For every $\mathfrak p ...
1
vote
1answer
83 views

Questions about a proof in Greenberg's Book.

I am trying to understand the proof of the following lemma : Lemma ' : Suppose that $X$ is a finitely generated $\Lambda$-module ($\Lambda =\mathbb Z_p[[T]]$) and that ...
1
vote
1answer
129 views

if $f \in A[x]$ is a zero divisor, then there exists $a ≠ 0$ in $A$ such that $af = 0$. [duplicate]

The title of the question indicates what I am attempting to prove, that if $f$ is a member of a polynomial ring over a commutative ring with identity, and $f$ is a zero divisor, then there exists a ...
1
vote
1answer
79 views

Behaviour of Betti tables with exact sequences

Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of finitely generated graded $S$-modules, where $S=k[x_1, \ldots, x_n]$ is a polynomial ring in $n$ variables. Let $\beta_{i,j}(M)$ denote the ...
2
votes
1answer
73 views

Homomorphism of modules and Tensor Product.

Let $\phi: A \rightarrow B$ be a ring homomorphism. Let $M$ be an $A$-module. We can think $B$ as $A$-module via the map $\phi$ defined by $\phi:A\times B \rightarrow B$, $(a,b)\mapsto\phi(a)\cdot ...
3
votes
1answer
114 views

Is the derived category of a commutative ring monoidal?

Let $A$ be a commutative ring, and consider the derived category $D(A)$. Is this a symmetric monoidal category? We have an obvious product, that is $-\otimes^L_A - $, and it is clear that we have an ...
2
votes
1answer
75 views

Integral Galois Extension (Serge Lang)

I have two questions about the proof of the following Proposition: Let $A$ be a ring, integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$ with group $G$. Let $P$ ...
3
votes
1answer
120 views

vanishing of Tor and regular sequences

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Let $x=x_1,\dots,x_n$ be an $R$-sequence such that it is also an $M$-sequence and let $I=(x_1,\dots,x_n)$. Question: Is it true that ...
3
votes
0answers
64 views

Irreducibility of $x^p - y^q$ in $K[[x,y]]$, for p,q>1 relatively prime

For $p,q>1$, relatively prime, $x^p - y^q$ is irreducible in $K[x,y]$. Is it also irreducible in $K[[x,y]]$ and how would you show it? I'm quite stuck at the moment. Also $K[x,y]/(x^p - y^q)$ is ...
0
votes
1answer
57 views

on the proof of a simple inequality in dimension theory

Let $(R,m)$ be a local ring and $M \neq 0$ a finite $R$-module. Let $x \in m$ and set $\bar{M}=M/xM$. Then $\dim M/xM \ge \dim M -1$. One way to see this is as follows: let $\dim M/xM = s$ and let ...
11
votes
4answers
160 views

Why is $\operatorname{Hom}(M,N)$ not necessarily an $R$ module?

Let $R$ be a ring, and $M,N$ be left $R-$modules. Then is it not true that $Hom_R(M,N)$ has the structure of an $R$-module? I was reading the preface of the Homological Algebra book by Rotman and ...
3
votes
2answers
152 views

Why is the following homomorphism not flat?

Let k be a field of characteristic 2. Consider the map $k[x,y]/(y^2+x^3) \rightarrow k[x,y]/(y^2+x^3)$ given by $x \rightarrow x^2$ and $y \rightarrow y^2$. Why is it that this map is not flat? I have ...
1
vote
1answer
67 views

why are these rings called fibres?

This question is self-contained. In the book "Monomial Ideals", by Herzog and Hibi, p. 45, we have the following definition: Definition: Let $K$ be a field. A one-parameter flat family of ...
2
votes
1answer
150 views

Is there an Noetherian ring (commutative) with exactly three prime ideals?

Is there an Noetherian ring (commutative) with exactly three prime ideals $P_i$ which satisfies the following statements? $P_1P_2=0$ and $P_3P_3=0$ $P_1P_3\neq 0$ and $P_2P_3\neq 0$
2
votes
5answers
194 views

An element does not belong to an ideal

How can I prove that the element $x-5$ does not belong to the ideal $(x^2-25,-4x+20)$ in $\mathbb Z[x]$. I tried to show that by proving $x-5\neq(x^2-25)f(x)+(-4x+20)g(x)$ for all $f,g$. Any ...
2
votes
1answer
360 views

Projective equivalence

Definition Two projective plane curves $F$ and $G$ are projectively equivalent if there is a $\varphi_A\in PGL_2(k)$ such that $F(x,y,z)=G(a_{00}x+a_{01}y+a_{02}z,a_{10}x+a_{11}y+a_{12}z, ...
0
votes
4answers
95 views

Localization with maximal ideal

Let R be local ring with maximal ideal P. Show that every element of R\P is invertible. Now let e be an element of R satisfying $e^2=e$. How can we prove $e \in {0,1}$.
2
votes
0answers
256 views

Normalization of a curve

This is a point in an exercise given during my Commutative Algebra course. Let $k$ be an algebraically closed field with characteristic different from $2$. Let $R=\frac{k[x,y]}{(y^{2}-f(x))}$, where ...
1
vote
1answer
39 views

Question about a paper, polynomials preserving congruence

In the paper, Interpolation Domains (Here), the beginning of the paper says: Let $K$ be a field ... The same does not hold for a domain $D$ (which is not a field), as polynomials in $D[X]$ ...
2
votes
1answer
63 views

An example of a $P$-primary ideal $I$ satisfying $I^2 = IP$

Give some examples of a $P$-primary ideal $I \not=P $ in a noetherian domain $R$ such that $I^2=PI $.
3
votes
0answers
70 views

Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
2
votes
0answers
76 views

Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
1
vote
1answer
130 views

P-primary Monomial Ideal

Let $P=(x_{1},...,x_{r})\subset S=K[x_{1},...,x_{n}].$ Show that a monomial ideal $Q$ is $P$-primary if and only if there exists a monomial ideal $Q'\subset T=K[x_{1},...,x_{r}]$ such that ...
0
votes
1answer
81 views

Nilradical strictly smaller than Jacobson radical.

In a preparation question for an exam, I am asked to give an example of a ring $A$ such that the nilradical $\operatorname{Nil}(A)$ is strictly smaller that the Jacobson radical $J(A)$. Here's how I ...
3
votes
1answer
183 views

where is the mistake in this “paradox”?

In the middle of page 34 in Bruns and Herzog, Cohen-Macaulay Rings, the authors present the following situation: Let $k$ be a field and let $R=k[X,Y]$ be a graded ring with grading induced by ...