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

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0
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1answer
161 views

Discrete Valuation Rings problem 2

An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying: i) $\phi(a) = \infty$ if and only if $a=0$. ii) $\phi(ab) = \phi(a) + \phi(b)$. iii) ...
6
votes
1answer
81 views

algebraic distance of an element of a ring from an ideal

Let $A$ be a commutative ring and $I$ an ideal. Does there exist a notion of "distance" of an element $x \in A$ from the ideal $I$? This "distance", need not be of the form $A\rightarrow \mathbb{R}$; ...
2
votes
2answers
223 views

Kernel of $p$-adic logarithm.

I'm completely clueless as to how to answer the following question: Let $K$ be a field of characteristic zero which is complete with respect to a non-Archimedean aboslute value $|\cdot|$. Let ...
0
votes
1answer
113 views

Ring of fractions problem

How do I can determine the ring of fractions of $\mathbb{Z}[X]$ ? I don't know the process that I have to follow for do it.
2
votes
0answers
100 views

Finitely many prime ideals lying over the same prime ideal [duplicate]

Let $A \subseteq B$ an extension of rings such that $B$ is an $A$-module finitely generated. Show that for every prime ideal $\mathfrak{p} \subseteq A$ there is only a finite number of prime ideals ...
0
votes
1answer
139 views

Hilbert Theorem of zeros

Use the Hilbert Nullstellensatz Theorem to prove the following result: Given $F_1, F_2, F_3 \in \mathbb{C} [X_1,\dots,X_n]$ polynomials checking the following conditions: $F_1$ is ...
2
votes
1answer
407 views

Commutative ring with unity Proof on the set of units?

the question is as follows (TRUE or FALSE.) If R is a commutative ring with unity, then the set of units in R forms a subring. (If true, give a short proof. If false, give a specic counter-example.) ...
6
votes
1answer
124 views

Embedding of free $R$-algebras

Let $R$ be any nontrivial commutative unital ring and $I$ and $J$ any sets with $|I|>|J|$. Does there exist an embedding of $R$-algebras $R[x_i; i\in I]\longrightarrow R[y_j;j\in J]$? When ...
2
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0answers
25 views

Analytical Independence

I am aware of the definition of analytical independence in Noetherian rings. I am wondering if anyone knows of any generalization of the concept (or similar concept ) to non-noetherian rings.
2
votes
1answer
174 views

Pure Submodules and Finitely Presented versus Finitely Generated Submodules

Let $A$ be a ring $M$ an $A$-module and $N$ a submodule. Definition: $N$ is called a pure submodule of $M$ if the sequence $0 \rightarrow N \otimes E \rightarrow M \otimes E$ is exact for every ...
3
votes
1answer
112 views

Finitely generated torsion module over a PID.

Let $A$ be a PID, $K$ be the field of fractions of $A$, and $M$ be a finitely generated torsion $A$-module. Let $M'=\text{Hom}(M,K/A)$ and $M''=\text{Hom}(M',K/A)$. I want to show that the evaluation ...
6
votes
1answer
109 views

What is the image of the map $\hom(V,V) \to \hom(\wedge^k V,\wedge^k V)$?

The title says it all. For the uninitiated: Any map $f:V \to W$ induces a map $\wedge^k V \to \wedge^k W$ by $v_1 \wedge \cdots \wedge v_k \mapsto f(v_1)\wedge \cdots \wedge f(v_k)$, so $\wedge^k(-)$ ...
10
votes
3answers
296 views

Number of prime ideals of a ring

Could anyone tell me how to find the number of distinct prime ideals of the ring $$\mathbb{Q}[x]/\langle x^m-1\rangle,$$ where $m$ is a positive integer say $4$, or $5$? What result/results I need to ...
5
votes
2answers
140 views

How to see that $\ker\left((X,Y)\otimes_R(X,Y)\to(X,Y)^2\right)\simeq k$ in $R=k[X,Y]$?

Let $k$ be a field, $R=k[X,Y]$ and $I=(X,Y)$, so that $R/I\simeq k$. I proved, using a projective resolution of $k$, that $\text{Tor}^R_2(k,k)= k$. I also proved that in general $$ ...
1
vote
1answer
230 views

Deduce that a Noetherian valuation ring is either a field or a Discrete Valuation Ring.

I'm trying to solve this question from a book and I have already proved 1. Let $R$ be a local domain which is not a field. Suppose that the maximal ideal $M$ of $R$ is principal and satisfies ...
2
votes
0answers
91 views

Integral dependence and fraction fields [duplicate]

Consider $\mathbb{Q}[x]\subset\mathbb{Q}(x)\subset\mathbb{Q}(x)[y]=:K$, where $$y^2=x,$$ and let $O_K$ be the integral closure of $\mathbb{Q}[x]$ in $\mathbb{Q}(x)[y]$. Show that ...
5
votes
2answers
291 views

Integral domains such that all proper factor rings are finite

Let $\mathbb Z$ be the ring of rational integers. If $a\in\mathbb Z$ is a non-zero element, then the factor ring $\mathbb Z/(a)$ is finite and has order $|a|$. If $\mathbb Z[i]$ is the ring of ...
7
votes
2answers
187 views

Nullstellensatz and the Fundamental Theorem of Algebra

I came across an interesting problem that basically said something along the lines of ``Show that Hilbert's Nullstellensatz is equivalent to the Fundamental Theorem of Algebra.'' My algebraic geometry ...
4
votes
1answer
97 views

quadratic extension of $\mathbb{Q}(X)$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the quadratic extension of ...
-1
votes
1answer
151 views

Algebraic Curves

Let $F$ be a non-constant polynomial in $k[X_1,...,X_n]$, $k$ algebraically closed. Show that $\mathbb A^n \setminus \mathrm{V}(F)$ is infinite if $n\geq 1$, and $\mathrm{V}(F)$ in infinite if ...
2
votes
1answer
72 views

Local rings and classifying singularities

My query is a little vague, but I'll try to be as concrete as possible. Is there some sense in which the local ring of an algebraic variety (or more general complex space) at a point depends only on ...
-1
votes
1answer
122 views

List of examples of commutative rings [closed]

For curiosity: Can anyone present the currently known list of examples of commutative rings? As Wikipedia says, one may include polynomial rings, rings of algebraic integers and p-adic integers. What ...
5
votes
3answers
191 views

Integral closure of $\mathbb{Q}[X]$ in $\mathbb{Q}(X)[Y]$

Consider the ring $\mathbb{Q}[X]$ of polynomials in $X$ with coefficients in the field of rational numbers. Consider the quotient field $\mathbb{Q}(X)$ and let $K$ be the finite extension of ...
0
votes
1answer
51 views

Kernel of canonical morphism in inductive limit (proof by induction)

Let $\langle I, \leqslant \rangle$ be a directed poset and $\langle M_i, \mu_{i,j} \rangle$ be a directed system of $A$-modules over $I$. Now let $$ C = \bigoplus\limits_{i \in I} M_{i}, $$ and $D$ ...
4
votes
1answer
608 views

Jacobson radical equal to nilradical in $R[X]$

Let $R$ be a non-zero commutative ring with identity. Let $\textrm{nilrad}(R)$ be the nilradical of $R$, which can be characterised either as the intersection of all prime ideals of $R$, or as the ...
6
votes
1answer
114 views

Injective hull and some Hom

Let $R$ be a commutative ring with unit. Suppose $P\in Spec(R)$ and let $E=E(R/P)$ be the injective hull of $R/P$. What can we say about $Hom_R(R/P, E)$. We know that $R/m\cong Hom_R(R/m, E)$, where ...
-1
votes
1answer
85 views

Commutative Algebra: nilpotent elements.

Let $f=\sum_{n=0}^{\infty}a_nx^n$. If $f$ and $a_0$ is nilpotent how I can prove that $f-a_0$ is nilpotent? Or if $f^n=0$ and $a_0^n=0$ how can I prove that $(f-a_0)^n=0$, where $n\in \mathbb N$.
1
vote
1answer
202 views

Show that a map is continuous in the Zariski topology

Let $R,S$ be two commutative rings with unity and $\alpha :R\to S$ be a ring homomorphism, for $f\in R$ is a non nilpotent element let $R_f$ denote the localization of $R$ with respect to the ...
6
votes
2answers
230 views

Submodules of a free module over a commutative ring

Let $R$ be a commutative unital ring, $I$ a set, and $R^{(I)}$ the free module on $I$. Can there be a submodule $R^{(J)}\cong M\leq R^{(I)}$ with $|J|\!>\!|I|$? Can $R^{(I)}$ be generated (as a ...
2
votes
1answer
150 views

Spec R is irreducible

A topological space is called reducible if $X=X_1\cup X_2$ for two closed subsets $X_1,X_2$ with $X_1\ne X\ne X_2$. Otherwise its called irreducible, want to show that $\text{Spec}(R)$ is irreducible ...
1
vote
0answers
39 views

Conntectedness of SpecR [duplicate]

Let $R$ be a commutative ring with unity, $e\in R$ is called and idempotent if $e^2=e$ and if $e\notin \{0,1\}$ then it is called a non-trivial idempotent.want to show that $\text{Spec}R$ is not ...
0
votes
1answer
87 views

Monomial ordering problem

I've got the following problem: Let $\gamma$, $\delta$ $\in$ $\mathbb R_{> 0}$. The binary relation $\preceq$ on monomials in $X,Y$ is defined: $X^{m}Y^{n} \preceq X^{p}Y^{q}$ if and only if ...
5
votes
1answer
80 views

hypersurface intersected with generic line

Let $f \in \mathbb{R}[x_1,\cdots,x_m]$ be a homogeneous multivariate polynomial of degree $n$. Now, for $u,v \in \mathbb{R}^n$ the form $f(\lambda u + \mu v)$ can be written as $\mu^n h(\lambda/\mu)$, ...
1
vote
1answer
82 views

Flatness of Formal Power Series

According to Matsumura's Commutative Ring Theory, Ex. 7.4, the direct product of flat modules over a Noetherian ring $A$ is flat. How can we use this result to conclude that the formal power series in ...
2
votes
1answer
271 views

Direct Sum/Product of Flat Modules

Let $\left\{M_{\lambda}\right\}$ be a family of flat $A$-modules and define $M = \bigoplus_{\lambda} M_{\lambda}$. Let $0 \rightarrow N' \rightarrow N$ be an exact sequence of $A$-modules. Tensoring ...
1
vote
1answer
66 views

Localization of a ring which is not a domain

Let $A$ be a ring (commutative with $1$), let $S$ be a multiplicatively closed subset of $A$, i.e $S$ is contained in $A$ , $1\in S$ and $a,b\in S$ implies $ab\in S$, for every $a,b\in A$. Consider ...
2
votes
1answer
182 views

localization of a module and annihilators

I've just started reading on my own about localizations of modules. I've run into a difficulty as follows: Let $R$ be a commutative ring with unity, $M$ an $R$-module, $S\subseteq R$ a multiplicative ...
16
votes
3answers
706 views

If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent

I'm solving a problem from Atiyah-Macdonald. I have to show that if $X=\mathop{\mathrm{Spec}}A$ is not connected then $A$ contains idempotents $e \neq 0,1$. The converse is easy. If $e \in A$ ...
3
votes
1answer
108 views

Computing kernel of ring homomorphism

I am trying to answer the question already asked here. My question is two parts: 1) I think I have found a proof on my own, could someone check it is valid? Modulo that ideal, $x_i\equiv a_i$ so ...
2
votes
1answer
105 views

When is a quotient of an $R$-module $E$ a submodule of $E$?

Let $R$ be a commutative ring with $1$. Suppose we are given a surjective $R$-module map $\varphi:E \to M \simeq E/N$. Are there any sufficient and/or necessary conditions for having an injective map ...
5
votes
3answers
235 views

Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$

I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z_1, y-z_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which ...
4
votes
1answer
164 views

commutative diagram

Let $$\begin{array} AA & \stackrel{f}{\longrightarrow} & B \\ \downarrow{h} & & \downarrow{h'} \\ C & \stackrel{g}{\longrightarrow} & D \end{array} $$ be a commutative ...
1
vote
1answer
58 views

A problem on power series

Let $P(T) = \frac { 1 + T + T^2 + \cdots + T^m}{ 1 - T^2( 1 + T + T^2 + \cdots + T^n)} = \sum _{i = 0}^\infty \beta_n T^n$ be a formal power series expansion. This kind of series arose while I was ...
9
votes
1answer
245 views

Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$

Let $A$ be a commutative ring with identity and, $\mathfrak{a}$ and $\mathfrak{b}$ ideals.I'm trying to find sufficient and necessary conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + ...
2
votes
1answer
110 views

Injectivity is a local property

Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How can I show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$ , then $M$ is ...
3
votes
2answers
225 views

Nilpotency of Maximal Ideal of Local Ring

What are the implications of the maximal ideal of a local ring $(A,m,k)$ being nilpotent? For example, $A$ is Artinian if and only if it is Noetherian. Any other interesting implications?
1
vote
1answer
68 views

Semicubical parabola coordinate ring

Let $C=\{(x,y)\in\mathbb{A}^2 \colon y^2-x^3=0\}$. Let $k[C]=k[x,y]/(y^2-x^3)$ be the coordinate ring of $C$. I read in a book that $k[C]$ is non a UFD. Is there a not too difficult and deep way to ...
4
votes
1answer
151 views

Exercise 4.30 of Eisenbud

I am doing exercise 4.30 from Eisenbud's Commutative Algebra With A View Towards Algebraic Geometry which I append here: Exercise 4.30: Suppose $k$ is a Noetherian ring and for every finitely ...
4
votes
1answer
115 views

$M$ is $\bigcap \operatorname{Ass}(M)$-primary

Let $R$ be noetherian ring and $M$ an $R$-module such that $\operatorname{Ass}(M)$ is a finite set. Prove that $M$ is $\mathfrak{b}$-primary, where $\mathfrak{b}=\bigcap ...
3
votes
0answers
131 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...