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

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2
votes
2answers
89 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 $I+J$...
3
votes
0answers
48 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 ...
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, ...
2
votes
2answers
104 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
votes
1answer
28 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
87 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
votes
0answers
94 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 ...
0
votes
1answer
28 views

A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. [closed]

I expect that the following result is true, but i can't prove it. A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. I need some help to prove this....
3
votes
1answer
40 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
vote
0answers
26 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
votes
1answer
56 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
votes
1answer
50 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 ...
-3
votes
1answer
66 views

Help finding an article [closed]

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
votes
2answers
79 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
66 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 $A$-...
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?
0
votes
1answer
47 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{...
1
vote
0answers
38 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
votes
0answers
32 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
121 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
vote
2answers
45 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 ...
1
vote
2answers
66 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
179 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
votes
0answers
59 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 ...
0
votes
0answers
56 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 ...
-1
votes
1answer
32 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
113 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
85 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
votes
1answer
106 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
vote
2answers
58 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
votes
0answers
28 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'+...
-1
votes
1answer
33 views

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

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
49 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
63 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
34 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
53 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
99 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{ ...
3
votes
1answer
74 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 ...
0
votes
0answers
42 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
votes
0answers
35 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
votes
1answer
53 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
36 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
72 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 $...
0
votes
0answers
48 views

Jacobson radical of formal power series ring

If $f=\sum_{i=0}^{\infty} a_i x^i \in R[[x]]$, let $\mathfrak{R}$ denote the Jacobson radical of a ring. I wish to show that $f\in\mathfrak{R}(R[[x]])\iff a_0\in\mathfrak{R}(R)$. I have already proved ...
4
votes
0answers
55 views

What are “generalized bases” really called, and where can I learn more?

(Notation: $f \diamond g$ means the composite $g \circ f$.) The following situation occurs frequently: We have an $\mathbb{R}$-algebra $A$, together with a distinguished set $I$ (the "indexing set"),...
0
votes
1answer
57 views

Compute the projective dimension of the given $R$-module

Let $$R=\frac{K[[x,y,z]]}{\left<xz,yz\right>}\text{ and } M=\frac{R}{\left<z+\left<xz,yz\right>\right>}.$$ Compute the projective dimension of $M$ as an $R$-module. My attempt ...
0
votes
1answer
61 views

If $A$ is an integral domain with a finite number of primes then $Q(A)=A_a$ for some $a \in A$ [closed]

If $A$ is an integral domain with a finite number of prime ideals is it possible to get the field of fractions localizing only by a set $\{a^k\}$?
0
votes
0answers
43 views

Global dimension of power series ring $k[[x_{1}, \cdots, x_{n}]]$

Let $R$ be the power series ring $k[[x_{1}, \cdots, x_{n}]]$ over a field $k$. Notice that $R$ is a noetherian local ring with residue field $k$. Show that $gl. \dim(R)=pd_{R}(k)=n$. By First Change ...
0
votes
1answer
34 views

Total quotient ring of $\mathbb Z_{2^n}$ [closed]

I want to characterize the total classical quotient rings of the (commutative) rings $R=\mathbb Z_4$, $R=\mathbb Z_8$ or any $R=\mathbb Z_{2^n}$. In fact, if we get $S$ to be the regular elements ...
-1
votes
2answers
69 views

If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$ [duplicate]

If $I$ and $J$ are ideals in a ring $R$ with 1 which are co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$ Work done: Should I proceed using Zorn'...