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

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4
votes
2answers
43 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
68 views

How to decide if a polynomial is symmetric?

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
votes
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 ...
1
vote
2answers
66 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 ...
2
votes
1answer
94 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 ...
-2
votes
1answer
26 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+...
1
vote
0answers
31 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 ...
1
vote
0answers
28 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.
0
votes
1answer
31 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
vote
1answer
37 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 ...
1
vote
1answer
37 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,...
0
votes
1answer
43 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
votes
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
votes
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
vote
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
votes
3answers
98 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 ...
1
vote
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 ...
4
votes
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
vote
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 ...
-2
votes
0answers
48 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
votes
1answer
56 views

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

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
votes
2answers
84 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
votes
1answer
84 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 ...
9
votes
3answers
762 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
61 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 ...
0
votes
0answers
24 views

What properties are preserved by direct limits? [on hold]

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 ...
3
votes
0answers
31 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 affine scheme $X=Spec A$. Let $Y = ...
1
vote
2answers
115 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
votes
1answer
84 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
votes
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.
0
votes
2answers
75 views

How to decompose that ideal? [on hold]

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
505 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 $...
0
votes
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
296 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
24 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
36 views

Prime spectrum of tensor product of two R-algebras [on hold]

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}(...
4
votes
0answers
84 views

Injectivity of $R \to R[t]/(f)$ for non-constant $f\in R[t]$

Question: Let $R$ be a (unital commutative) ring and $f = a_0 t^n + \cdots + a_n \in R[t]$ a non-constant polynomial. What are (necessary and sufficient) conditions on the coefficients $a_0,\ldots,a_n ...
3
votes
1answer
37 views

Zariski tangent vectors, dual numbers

Let $k$ be a field, $A$ be a Noetherian local $k$-algebra, $m$ its maximal ideal, and an isomorphism $i:A/m \to k$ . Let $v:m/m^2 \to k$ be a $k$-linear map (i.e. a Zariski tangent vector). I believe ...
29
votes
6answers
2k views

Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$

This seems to be one of those tricky examples. I only know one proof which is quite complicated and follows by localizing $\mathbb{Z}[\sqrt[3]{2}]$ at different primes and then showing it's a DVR. ...
1
vote
0answers
24 views

Example of an monomial ideal that is weakly reverse lexicographic but not reverse lexicographic

We are looking at a paper titled "Generic Ideals and Moreno-Socias Conjecture" by Edith Aguirre, et al. In the paper they state that an ideal which is reverse lexicographic is also weakly ...
1
vote
2answers
209 views

About second uniqueness primary decomposition theorem

I'm self-learning commutative algebra from Introduction to Commutative Algebra of Atiyah and Macdonald and get frustrated about the second uniqueness primary decomposition theorem. I copy the theorem ...
0
votes
1answer
49 views

How do you find the free resolution of the module $M$ and of $F/M$ where $F=(K[x,y])^3$?

$M$ is a module generated by $$f_1=(xy,y,x), f_2=(x^2+x,y+x^2,y), f_3=(-y,x,y),f_4=(x^2,x,y).$$ We're to use the lex ordering with $x<y$ and $e_1>e_2>e_3$, where terms are given preference ...
0
votes
1answer
49 views

Finite type + integral = finite

Let $A \subseteq B$ be rings (comm. with unity). I am struggling to see why the following equivalence holds for $B$ interpreted as a $A$-Algebra: $A \rightarrow B$ is of finite type and $A\...
1
vote
1answer
44 views

Intersection of flat submodule with direct summand

Let $R$ be a (commutative) domain, $M$ a flat $R$-module which decomposes as $M=A\oplus B$ and $N$ a (not necessarily pure) flat submodule of $M$. Is it the case that $N \cap A$ is always a pure ...
0
votes
1answer
85 views

How to compute $\dim_{\mathbb C}\mathbb{C}[x,y,z]/(z^4,x^2+y^2+z^2-1,xy)$?

How to compute $\dim_{\mathbb C}\mathbb{C}[x,y,z]/(z^4,x^2+y^2+z^2-1,xy)$? I tried to decompose $$(z^4,x^2+y^2+z^2-1,xy)=(z^4,x^2+y^2+z^2-1,x)\cap(z^4,x^2+y^2+z^2-1,y)=(z^4,x^2+z^2-1,y)\cap(z^4,y^2+...
0
votes
1answer
36 views

If some polynomial is in an ideal $I$, how can I write it as a linear combination of the generators of $I$?

I'm looking for a (easy) procedure of some sort. I also know a little bit of Singular and CoCoA, and was wondering if you can do that in there?
1
vote
1answer
64 views

Extension of Scalars is well-defined

The reason I'm asking this, is because as an exercise, I'm asked to prove the following: Let $A$, $B$ be rings, $f:A\to B$ a ring homomorphism inducing $A$-module structure on $B$, and $M$ a flat $A$-...
2
votes
2answers
77 views

Quotient of a polynomial ring localized

Question: Prove that $\mathbb{R}[x,y]/(xy)$ localised at $(x-a)$ is isomorphic to the ring $\mathbb{R}[x]$ localised at $(x-a)$. Related question: What is the local ring at the point $(0,0)...