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

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1answer
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Modules as morphisms to endomorphism rings

An $A$-module $M$ may be thought of as a (surjective) ring homomorphism $f: A \to E(M)$, where $E(M)$ is a ring of group endomorphisms of $M$. Then $am = f(a)(m)$. Is there any more to this ...
2
votes
1answer
34 views

Irreducible elements for a commutative ring that is not ID

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 ...
3
votes
3answers
45 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 ...
2
votes
2answers
68 views

How do I find the ideal $I+J$?

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 $I+J$...
0
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0answers
21 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 ...
2
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0answers
29 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 = ...
0
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0answers
14 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
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1answer
48 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 that isomorphism. Does it hold? Thank you.
1
vote
1answer
45 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 ...
0
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0answers
33 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}(...
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 ...
2
votes
1answer
46 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 ...
4
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0answers
79 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
35 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 ...
1
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0answers
23 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 ...
0
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1answer
46 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\...
0
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1answer
47 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 ...
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1answer
63 views

Help finding an article [on hold]

Hello Recently I have been studying algebra and am in search of the following paper : Kac, V. G. Classification of simple $Z$-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra 5 ...
0
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2answers
70 views

How to decompose that ideal?

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 ...
1
vote
1answer
61 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 $A$-...
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0answers
23 views

Checking regularity via completion

It is well-known that a local Noetherian ring $A$ is regular if and only if its completion is regular, and that one can check (if, say, $A$ is a $k$-algebra) by observing that it is a power series ...
0
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1answer
34 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?
0
votes
1answer
43 views

Definition of singular points on an algebraic curve

From what I understood, given a point $p$ on a scheme $X$ over a field $k$, we have \begin{equation} \dim \mathcal{O}_{X,p} \leq \dim_{\mathcal{O}_{X,p}/\mathfrak{m} }\mathfrak{m}/\mathfrak{m}^2 \end{...
0
votes
0answers
31 views

Structure constants in a finitely generated $\mathbb{k}$-algebra

Let $\mathbb{k}$ be a field of characteristic $0$. Suppose we have a finitely generated graded $\mathbb{k}$-algebra $A= \bigoplus_{i=0}^{\infty}A_i$ which is free of finite rank as a module over a ...
0
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0answers
28 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
1
vote
2answers
113 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 ...
1
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2answers
41 views

Is an algebra homomorphism between two finitely generated algebras over a field automatically an integral morphism?

I'm having a bit of trouble with the idea of an integral morphisms, and algebra homomorphisms for that matter. I'm wondering if the above is just "automatically" true. Does an algebra over a field ...
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0answers
40 views

relation between the Krull dimension and the dimension of vector spaces

Let $(R, \frak m)$ be a commutative Noetherian local ring, and $M$ be an $R$-module such that $\frak m$$M =0$. We know that $M$ can be considered as a $R/ \frak m$-module, namely as a vector space on ...
1
vote
2answers
64 views

If $R\otimes_\mathbb R\mathbb C$ is finitely generated $\mathbb C$ - algebra then $R$ is a finitely generated $\mathbb R$ - algebra?

Let $R$ be an $\mathbb R$ - algebra. Suppose $A=R\otimes_\mathbb R\mathbb C$ is a finitely generated $\mathbb C$ - algebra then is $R$ a finitely generated $\mathbb R$ - algebra? I thought along the ...
6
votes
4answers
152 views

polynomial ring with isomorphic quotients

If $R$ is a commutative ring and $f(x), g(x) \in R[x]$ two polynomials such that $R[x]/f(x)\cong R[x]/g(x)$ as $R$-algebras, what can we say about $f$ and $g$? Or given $f(x)\in R[x]$, what can we ...
0
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0answers
55 views

Dimension of polynomial rings and tensor products of residue fields

In Matsumura textbook to show that $\dim A[x] = \dim A + 1$, first it states that $A[x] \otimes k(\mathfrak{p}) = k(\mathfrak{p})[x]$ which is one dimensional. Then it uses the theorem 15.1.(ii) since ...
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0answers
52 views

primary decomposition of injective envelope of a module

The Exercise A3.6 of Eisenbud's book, Commutative Algebra with a view Toward Algebraic Geometry, is: Assuming that $R$ is Noetherian, let $M$ be any finitely generated $R$-module. a. Let ...
0
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1answer
30 views

How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$?

How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$? I know the answer in the sense that i know that $(2x+1)$ is a maximal principal ideal of that polynomial ring. But if i ...
5
votes
1answer
111 views

Is there a commutative ring with a “generalized determinant”?

Does there exist a commutative ring(-with-a-1) $R$ and positive integer $n$ and function $\hspace{.04 in}f$ from [the set of $n$-by-$n$ matrices over $R$] to $R$ such that $f$ is linear in each row ...
0
votes
1answer
80 views

Tensor products and Residue fields

Given a ring homomorphism between two Noetherian rings, $f:A \to B$. Let $P$ be a prime ideal in $B$ and let $\mathfrak{p}$ be an ideal in $A$ such that $f^{-1}(P) = \mathfrak{p}$. How can we prove ...
0
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1answer
99 views

Krull dimension of three modules [closed]

Let $R$ be a commutative Noetherian ring with non-zero identity and $M$, $M'$ and $M''$ be $R$-modules (not necessarily finitely generated) with $\operatorname{Supp} M \subseteq‎ \operatorname{Supp} M'...
1
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2answers
52 views

Residue field of the integral closure of a local ring in its field of fractions

When considering the discrete valuation rings contained in the rational functions field $R(F)$ of an irreducible plane projective curve $F \in \mathbb{P}^2(K)$ ($K$ algebraically closed), one can find ...
0
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0answers
26 views

Singular ideal of an idealization

Let $S$ be a commutative ring, and let $A$ be a faithful $S$-module. Through idealization, we can make the abelian group $R=S⊕A$ into a commutative ring using the multiplication $(s,a)(s',a')=(ss',sa'+...
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1answer
32 views

Localization at associated prime of a principal ideal and ideal generator [on hold]

Let $A$ be a commutative Noetherian local ring and $I=(a)$ a principal ideal of $A$. Let $P$ be an associated prime of $A/I$. Is $a$ a maximal regular sequence on $A_P$ (i.e., $a$ is not a zero ...
2
votes
1answer
47 views

$k\left[x,y\right]$ is not integral over the $k\left[xy,y\right]$

I want to prove that the polynomial ring $k\left[x,y\right]$ is not integral over the subring $k\left[xy,y\right]$ , where $k$ is a field. My claim is that $x$ is not integral over $k\left[xy,y\...
1
vote
1answer
57 views

Rings of Krull dimension one

I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate ...
0
votes
1answer
33 views

Induced homomorphism on Spectra of rings

In Matsumura textbook, there is this following statement. A ring homomorphism $f:A \to B$, induces a map $f': \operatorname{Spec}B \to\operatorname{Spec}A$ under which an element $\mathfrak{p} \...
1
vote
1answer
51 views

Behaviour of an étale morphism under Galois action on points.

Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such ...
5
votes
3answers
94 views

Let $R$ be a commutative ring, $\phi :R\to S^{-1}R, \phi(r)=\frac{r}{1}$ then $\phi(r)$ is invertible iff $r\in S$

$R$ is an arbitrary commutative ring with identity, and $S\subset R$ is multiplicative. I read that the map $\phi :R\to S^{-1}R, \phi(r)=\frac{r}{1}$ is characterized by the set $S'=\{s:\phi(s)\text{ ...
2
votes
0answers
56 views

On 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 ...
0
votes
0answers
41 views

Motivation for localization as given in Eisenbud

Eisenbud writes that the affine ring $A(X-Y)$ is obtained from $A(X)$ by adjoining a multiplicative inverse of $f$, where $Y$ is the vanishing set of the function $f$. $A(X-Y)$ is the set of ...
0
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0answers
34 views

grading of the tensor product

I have just had a look at http://therisingsea.org/notes/GradedModules.pdf to look up the grading of the tensor product of two graded modules over a graded ring (see page 10). And I am wondering, why ...
0
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1answer
52 views

Let $\phi:A\to B$ be a ring homomorphism, $\phi^{*}:Y\to X$ the induced continuous map on $X=\mathrm{Spec}(A), Y=\mathrm{Spec}(B)$.

This is from Atiyah and MacDonald, Exercise 1.21, part iii). We let $Z=\mathrm{Spec}(R)=\{\mathfrak{p}\subset R\mid\mathfrak{p}\mathrm{\,a\,prime \,ideal}\}$ have the Zariski topology, i.e. with ...
0
votes
1answer
35 views

A reduction to the finite degree case

I am stuck trying to understand a proof in Asymptotic Differential Algebra and Model Theory of Transseries by L. van den Dries, J. van der Hoeven and M. Aschenbrenner. The result is the following: ...
1
vote
0answers
71 views

If $R'$ is an $R$-algebra, $M,N$ are $R'$-modules, when do we have $M\otimes_{R}N\simeq M\otimes_{R'}N$ naturally?

Suppose $R$ is a commutative ring with unity, and $R'$ an $R$-algebra with structure map $\phi: R\to R'$. Let $M,N$ be two $R'$-modules. Then there exists a natural $R'$-linear (hence $R$-linear) map $...