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

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1answer
42 views

Decomposition of a monomial ideal

I have to find a primary decomposition of the following ideal and I proceeded in this way: $$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,...
1
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0answers
20 views

How can I compute completions of rings?

I want to learn about how to compute the completions of local rings. For example, I want to be able to compute the completions of \begin{align*} \left(\frac{\mathbb{C}[x,y]}{(y^2 - x)}\right)_{(x,y)} ...
5
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1answer
80 views

Castelnuovo-Mumford regularity and exact sequence.

In a question on MathOverflow it is said that: It is known that given a short exact sequence of finitely generated graded modules over a polynomial ring over a field:$$0 \to M'' \to M \to M' \to 0$...
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2answers
23 views

Isomorphism of modules [duplicate]

Are $\mathbb{C}[x,y]/(x,y)$ and $\mathbb{C}[x,y]/(x-1,y-1)$ isomorphic as $\mathbb{C}[x,y]$-modules? I think they are cyclic so they are isomorphic, but I'm not sure.
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2answers
90 views

Example of non-noetherian ring whose spectrum is noetherian and infinite

A topological space is noetherian if it satisfies the descending chain condition for its closed subsets. Let be $R$ a commutative ring and let $\mathrm{Spec}(R)$ its spectrum with Zariski topology. I ...
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0answers
20 views

Homogeneous System of Parameters

Assume that $R$ is a finitely generated graded $k$-algebra of Krull dimension $n$. Is it true that any set $\{f_{1},f_{2},...,f_{n}\}$ of homogeneous algebraically independent polynomials is a ...
3
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1answer
69 views

On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
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1answer
32 views

Faithfully flat descent of projectivity and freeness

I am reading this paper. It is proven there that if $f:A\rightarrow B$ is a faithfully flat morphism of rings and $M$ an $A$-module such that the $B$-module $M\otimes_A B$ is projective, then $M$ ...
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0answers
13 views

Freeness over a local ring.

Let $A$ be a (Noetherian) local ring and $M$ a finite-type $A$-module. In order for $M$ to be free does it suffice that $\mathrm{Ext}^1_A(M,k)=0$, where $k$ is the residue field? I would appreciate ...
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2answers
27 views

Maximal ideal in a local artinian ring.

I know that an artinian ring $A$ is the union of its units and its zero-divisors. So every non-zero-divisor is an unit. I also know that in a local ring every element which is out from the maximal ...
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1answer
13 views

A condition that the ratio of locations is maximal

Sea $R$ un anillo conmutativo con identidad e $I$ un ideal de $R$ y $m$ un ideal maximal de $R$. Mostrar que $\displaystyle\frac{R_m}{I_m}\neq{0}$ si y solo si $I\subseteq{m}$. Dm: $[\Rightarrow{}]$. ...
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0answers
30 views

Find an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(m\in M)$

Let $R$ be a commutative ring with $r_0\in R$ a fixed element, and $M$ be an $R$-module. I search for an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(...
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0answers
15 views

Combinatorial commutative algebra

Let G is simple graph and Δ(G) is clique complex, IΔ(G) has a 2-linear resolution if and only if, for any subset W ⊂ [n], one has H˜i(Δ(G)W ; K) = 0 unless i=0
3
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0answers
40 views

Is quotient of open invariant subset open?

I am reading GIT book by Mumford. He needs special cases of the following conjecture several times. Conjecture Let $G$ be a reductive algebraic group acting on an irreducible affine scheme $X=Spec ...
4
votes
1answer
209 views

Local ring with intersection of powers of its principal maximal ideal zero

In an algebra test the following problem was presented: Any commutative local ring $(R,\mathfrak m)$ with $\mathfrak m$ principal so that $⋂_{i≥0}\mathfrak m^i =0$ is Noetherian and each nonzero ...
5
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2answers
57 views

Finite commutative ring with unity and without nilpotent elements

Let $R$ be a commutative ring with unity such that for each $x \in R$ there exists a $n \in \mathbb{N}$, $n>1$, such that $x^n = x$. Then show that $$ R\simeq F_{1}\times F_{2}\times \cdots\times ...
2
votes
1answer
103 views

How to decide if a polynomial is symmetric? [on hold]

First, is the following: $$f=\frac{3}{5}(x_1^5 + x_2^5 + x_3^5 + x_4^5)-\frac{7}{12}(x_1^2x_2^2 - x_1^2x_3^2-x_1^2x_4^2-x_2^2x_3^2-x_2^2x_4^2-x_3^2x_4^2)$$ a symmetric polynomial? And, if yes, how do ...
3
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1answer
152 views

A ring with ACC on prime ideals whose spectrum is non-noetherian.

I am currently working on the converse of the exercise #12 on chapter 6 of Atiyah-Macdonald's book on commutative algebra. The problem is asking whether there is a ring $A$ which satisfies the ...
2
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1answer
96 views

Relation between stalks of twisted sheaf and structure sheaf

Let $A$ be a ring, $B = A[T_0,\dots, T_d]$, and $X = \textrm{Proj } B$. Then at every point $x \in X$, $$\mathcal{O}_X (n)_x \cong \mathcal{O}_{X,x}$$ Let $x$ correspond to a homogeneous prime ...
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1answer
31 views

about minimal prime ideals [on hold]

Let $R$ be a ring with minimal prime ideals $p_1,\ldots, p_n$ and $D=R/{p_1}\times \cdots \times R/p_n$. Please find an element $x\in R$ such that $\mathrm{ann}_D(x+p_1,\ldots,x+p_n)=\mathrm{ann}_D(1+...
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0answers
40 views

Hartshorne's algebraic geometry ; geometric understanding and intuition for intersection multiplicity

I am reading section $7$ of the book. He defines intersection multiplicity as Let $Y$ be a projective variety of dimension $r$. Let $H$ be a hypersurface not containing $Y$. Then by (7.2) $Y\cap ...
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0answers
33 views

Understanding the Definition of minimal prime ideal of a graded module

I am reading algebraic geometry from Robin Hartshorne. He has used a term "$p$ is a minimal prime of a graded $S$ module $M$". What does it mean? I know the definition of minimal prime over an ideal.
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1answer
33 views

Let $M_1$, $M_2$ be Artinian modules over $R$. Then $M_1\times M_2$ is Artinian.

Using exact sequences, it's fairly easy to prove the converse, but I can't figure out how to prove this statement. Suppose we have a descending chain $N_1\supset N_2\supset\cdots$ of $R$-submodules ...
1
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1answer
39 views

Noetherian vector space is finite-dimensional

Given a field $k$, and a $k$-vector space $V$ which is noetherian as $k$-module, I want to show that $V$ is finite-dimensional. Is it correct that this follows because since $V$ is noetherian, every ...
0
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1answer
48 views

The ideal of the image of homogeneous polynomials

Let $k$ be an algebraically closed field, and $f_0,\dots,f_m \in k[x_0,\dots,x_n]$ be homogeneous polynomials of the same degree. Denote by $I\subset k[x_0,\dots,x_m]$ the kernel of the homomorphism ...
5
votes
2answers
116 views

UFD yields height of certain primes at most $1$

Let $R$ be a unique factorization domain. If $P$ is a prime ideal minimal over a principal ideal, is it true that height of $P$ is at most $1$? In case $R$ is Noetherian the result follows due to ...
6
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3answers
1k views

Does $A$ a UFD imply that $A[T]$ is also a UFD?

I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD. The only thing I am sure I could try to use is Gauss's lemma. Also, how can we deduce that the polynomial rings $\mathbb{Z}[x_1,\...
4
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2answers
28 views

Noetherian semiprimary rings

Is any Noetherian semiprimary ring $R$ Artinian? By semiprimary I mean $R/J(R)$ semilocal and $J(R)$ nilpotent, where $J(R)$ is the Jacobson radical of $R$. I know that if $R$ is Artinian then $J(...
1
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1answer
46 views

I'm stuck with trying to construct a $K$-basis for the quotient of the polynomial ring $S/I$.

We were told in class that a $K$-basis for $S/I$ where $S=K[X_1, \dots , X_n]$ and $I$ a monomial ideal in $S$ is $W = \{X^a \in \mathrm{Mon}(S) \mid X^a \notin I\}$. I'm having difficulties ...
3
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3answers
101 views

Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
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0answers
24 views

$R$ integral domain, $P$ projective and injective module $\implies P=0$ or $R$ is field of fractions [duplicate]

I'm having difficulties proving the following: Let $R$ be an integral domain and $P$ a projective and injective $R$-module. Show that $P=0$ or $R=Q(R)$, where $Q(R)$ denotes the field of fractions of ...
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3answers
64 views

Studying the intersection $(X)\cap (X^{2}-Y+1)\subseteq\mathbb{R}[X,Y]$.

I am trying to find the intersection of ideals $$ (X)\cap (X^{2}-Y+1)\subseteq\mathbb{R}[X,Y]. $$ This is what I have tried: $$ f\in(X^{2}-Y+1)\Rightarrow f=g\cdot (X^{2}-Y+1)\text{ for certain }g\...
1
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1answer
63 views

Extension of ideals in integral extensions

Let $R\subset S$ be an integral extension in the category of commutative rings with unity. I have three questions: 1) Is every ideal of $S$ an extended ideal? 2) Is extension of each idempotent ideal ...
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0answers
49 views

Minimal prime ideal and sum of two ideals [on hold]

Let $R$ be a commutative ring with $1$, and let $p$ be a minimal prime ideal of $R$. If $p\subseteq I_1+ I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $ p\subseteq I_1 $ or $...
0
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1answer
57 views

Is it true that $\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x)$? [closed]

I need to show that $(xy^2-1)$ is prime in $\mathbb{Q}[x,y]$ and I tried to consider the isomorphism $$\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x).$$ Does it hold? Thank you.
2
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2answers
86 views

How do I find the ideal $I+J$ and quotient $R/(I+J)$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
2
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1answer
85 views

A prime ideal which is not maximal

I am searching for a prime ideal of the ring $R=∏_{n=2}^{∞} {\mathbb Z}_{2^n}$ which is not maximal. In fact, since each ${\mathbb Z}_{2^n}$ is local with $\left<\bar 2\right>$ as the maximal ...
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3answers
763 views

Is noetherianity a local property?

Let $R$ be a ring with finitely many maximal ideals such that $R_{\mathfrak m}$ ($\mathfrak m$ maximal ideal) is noetherian ring for all $\mathfrak m$. Is $R$ noetherian? I think $R$ has to be ...
2
votes
1answer
64 views

Irreducible elements for a commutative ring that is not an integral domain

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
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0answers
25 views

What properties are preserved by direct limits? [closed]

We know that direct limit of a directed family of flat $R$-modules is also flat ($R$ is a commutative ring with $1$ and all modules are unital). I am looking for other properties of modules which ...
1
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2answers
116 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
0
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1answer
86 views

Why is $(x,y)\cap(x,z)\cap(x,y,z)^2$ a minimal primary decomposition of $(x,y)(x,z)$?

Why is $(x,y)\cap(x,z)\cap(x,y,z)^2$ a minimal primary decomposition of $(x,y)(x,z)$? I understand that the ideals are primary and also that one has $$(x,y)\cap(x,z)\cap(x,y,z)^2=(x,y)(x,z).$$ But I ...
2
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1answer
51 views

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
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2answers
75 views

How to decompose that ideal? [closed]

We have $$I=\left(x^2+2y^2-3,y(x-y),y(y+1)(y-1)\right)\subset\mathbb{C}[x,y]$$ and I would like to decompose it as intersection of simpler ideals. How could I proceed? For example, in this ...
10
votes
2answers
506 views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
8
votes
5answers
1k views

Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all $...
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0answers
15 views

Prove that integral closure of $\mathbb R[x,y]/(y^2-x^3-x^2)$ is $\left( \mathbb R[x,y]/(y^2-x^3-x^2) \right) \left[ \frac{y}{x} \right]$ [duplicate]

i have to give a proof of the Headline. I just showed, that $y/x$ is integral over $R:=\mathbb R[x,y]/(y^2-x^3-x^2)$. How do I show, that $\bar R = R[t]$ where $t=y/x$? Furthermore, I have to show, ...
0
votes
2answers
297 views

Noetherian ring with infinite Krull dimension.

I just started to read about the Krull dimension (definition and basic theory), at first when I thought about the Krull dimension of a noetherian ring my idea was that it must be finite, however this ...
0
votes
1answer
25 views

In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
0
votes
0answers
37 views

Prime spectrum of tensor product of two R-algebras [closed]

Let $R$ be a commutative ring and $A_1$ and $A_2$ two commutative unital $R$-algebras. Is there any characterization for $\mathrm{Spec}(A_1\otimes_R A_2)$? Or how can we deduce that $ \mathrm{Spec}(...