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

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1answer
61 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
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2answers
276 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
60 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
244 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
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1answer
131 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 ...
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2answers
260 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 ...
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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
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1answer
82 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
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1answer
123 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
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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
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1answer
68 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
110 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
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1answer
72 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
119 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
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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
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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
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4answers
159 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
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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
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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
190 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
323 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
84 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
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0answers
230 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
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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
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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
69 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
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0answers
75 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
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1answer
128 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
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1answer
80 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 ...
3
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0answers
55 views

Algorithm for primary decomposition of ideals in a power series ring over a field

Let $K$ be a field such that there exists an algorithm for factoring a polynomial over $K$ into the product of irreducible polynomials. For example, the field of rational numbers $\mathbb{Q}$ is such ...
3
votes
1answer
115 views

Exercise 1.2.4 and Example 4.3.6 in Liu

I want to prove that if $X$ is a noetherian scheme then any flat closed immersion into $X$ is open, that is, if $A$ is noetherian then $\varphi:\operatorname{Spec}(A/I)\to\operatorname{Spec}(A)$ is ...
1
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1answer
103 views

graded modules have enough projectives

In the bottom of page 32 in Bruns and Herzog, Cohen-Macaulay Rings, the authors write that if $R$ is a $\mathbb{Z}$-graded ring and $M$ a $\mathbb{Z}$-graded $R$-module, then $M$ is the homomorphic ...
3
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0answers
95 views

Criterion for etaleness

Suppose $A \rightarrow B$ is an inclusion of commutative rings which is finite, so that $B$ is generated as an $A$-module by elements $ b_1, \ldots, b_n$ say. Are there any algebraic conditions on ...
1
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1answer
70 views

Short Exact Sequence of R-Modules & Chains

Let $L\xrightarrow{f}M\xrightarrow{g}N$ be a short exact sequence of $R$-modules, and assume that there is a chain of submodules $0=M_0<M_1<\dots <M_n=M$ in which the quotient ...
1
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0answers
79 views

Integral Farkas Lemma

The context of this question is commutative algebra, however the question itself is more related to convex geometry. All necessary information is given. In the proof of Lemma 3.1.1 in the book ...
1
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1answer
85 views

The order of the cokernel of an endomorphism over $ \mathbb Z_p$

I want to prove the following result : Let $X$ a finite-rank free $\mathbb{Z}_p$-module, and $\varphi \colon X \to X$ an endomorphism of $X$. Then $$|M/\varphi(X)| < \infty \Leftrightarrow ...
3
votes
1answer
48 views

Question from Cartan-Eilenberg, Chapter 6, exercise 5

The exercise problem is this; consider a unital ring $A$. For each right $A$-module $M$ and left ideal $I$ of $A$, TFAE. (a) For each relation $\:\sum _{i} a_iu_i=0 \:(a_i\in M, u_i\in I)$ there ...
2
votes
2answers
72 views

Non-unital commutative ring without non-prime ideals?

Does there exist a non-unital commutative ring such that all its proper ideals are prime? Note also that that equipping the abelian group $\mathbb Z/p\mathbb Z$ with trivial multiplication $xy=0$ for ...
1
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1answer
71 views

Tensor of quotients

I would like to prove the following: let $A$ be a ring and $I,J\subset A$ two ideals. Then: $$A/I\otimes_AA/J\cong A/(I+J)$$ I have seen a proof using the Yoneda lemma (even though I haven't ...
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1answer
70 views

Reference to this rule $I = (ab) + J$, then $I = ((a) + J) \cap ((b) + J)$

Could someone show the reference of this rule: if $I = (ab) + J$, then $I = ((a) + J) \cap ((b) + J)$. I have found it here.
1
vote
1answer
48 views

relation between projective dimension (pd) of ideal and pd of the quotient

Let $R$ be a Noetherian ring and $I$ a proper ideal of $R$. Suppose that the projective dimension of $I$ is equal to $n$. Let $0 \rightarrow P_n \rightarrow \cdots \rightarrow P_0 \rightarrow I ...
2
votes
1answer
68 views

Cox rings of toric varieties

Let $X$ be a projective toric variety over a field $k$. Is it true that the Cox ring of $X$ is the polynomial ring over the Picard group of $X$? If not, what is the significance of the Picard group ...
1
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1answer
104 views

An equivalence for $\operatorname{grade}(I,M)\ge 2$. [duplicate]

Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Show that $\operatorname{grade}(I,M)\ge 2$ iff the canonical homomorphism $M \mapsto\operatorname{Hom}_R(I,M)$ is an ...
1
vote
1answer
101 views

Prove that $A[x]$ is a flat $A$-algebra.

This is from exercice 5, chap 2 from Atiyah and McDonald "Introduction to Commutative Algebra". Let $A[x]$ be the ring of polynomials in one indeterminate over a ring $A$. Prove that $A[x]$ is a ...
2
votes
1answer
101 views

Viewing Laurent polynomials as a localization of $R[X]$?

I believe that the Laurent polynomials over a ring $R$ are simply the localization of $R[X]$ at $S=\{X^n:n\geq 0\}$. However, I've always thought as Laurent polynomials as like "polynomials" in that ...
4
votes
0answers
198 views

Gorenstein ring VS. Gorenstein singularity

A normal variety is said to have Gorenstein singularity iff its canonical divisor is a Cartier divisor (one can always define the canonical divisor on a normal variety and it can be proved to be a ...
3
votes
2answers
95 views

Localizations of $\mathbb{Z}/m\mathbb{Z}$

Let $A=\mathbb{Z}/m\mathbb{Z}$. Prove that for each multiplicative subset $\Sigma$ of $A$ there is an integer $n$ such that $\Sigma^{-1}A=\mathbb{Z}/n\mathbb{Z}$.