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

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3
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2answers
148 views

Quotient field of a certain quotient ring

Let $A$ be a commutative integral domain and $\mathfrak p$ a prime ideal of $A$. Let $A_{\mathfrak p}$ be the localization of $A$ at $\mathfrak p$ and $\mathfrak{m}_{\mathfrak{p}}=\mathfrak pA_{\...
6
votes
2answers
1k views

Rank of free submodules of a free module over a commutative ring. [duplicate]

Free modules over a commutative ring $R$ with $1$ have well-defined rank. I have been wondering if there is a ring $R$ such that there are free modules $M'\subset M$ with $\operatorname{rank}(M')&...
6
votes
1answer
139 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
2
votes
0answers
27 views

If an ideal contains a “toric” polynomial, does then also the Groebner basis contain such polynomial?

Suppose $k$ is an algebraically closed field, $I \subset k[x_1, \ldots, x_n]$ is an ideal that contains a polynomial of the form $x_1^{m_1} \cdots x_r^{m_r}+cx_{r+1}^{m_{r+1}}\cdots x_n^{m_n}$, where $...
0
votes
0answers
72 views

Minimal prime ideals are made of zero-divisors [duplicate]

Let $R$ be a commutative ring with unity which is not an integral domain. Let $P$ be any minimal prime ideal of $R$. How can I show that $P⊆Z(R)$, where $Z(R)$ denotes the set of zero-divisors of $R.$
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votes
0answers
161 views

Irreducible Elements, Units, UFD

Let $P$ be a set of positive prime numbers. Let $\mathbb{Z}_{P}$ be the collection of all rational numbers of the form $a/b$, where $a,b$ are integers, $b$ not in $0$, and for all $p \in P$, $p$ does ...
4
votes
1answer
107 views

Completion of a ring

Consider the ring $$R= \mathbb{Z}_p[x,y]/((x^2-2+y^2)(x^2-y^2)+p^ry),$$ where $p$ is an odd prime and $r$ is an integer greater than $1$. I want to show that the completion of $R$ at $(p,x,y)$ is ...
5
votes
1answer
197 views

Finitely generated ring with zero Krull dimension

I'm trying to prove the following: Every finitely generated ring with Krull dimension equal to zero is finite. I'm trying to show that the ring is a domain, hence a field, in order to use the ...
4
votes
1answer
111 views

DVRs Sitting in an Extension

Prompted by this question, I was wondering if the following had any simple solution. Definition: Let $L/K$ be any extension of fields. Define $D(L/K)$ to be the set of all DVRs $R$ such that $...
5
votes
1answer
158 views

The uniqueness of a special maximal ideal factorization

The following problem is from Michael Artin's Algebra, chapter 12, M.6, unstarred: Let $R$ be a domain, and let $I$ be an ideal that is a product of distinct maximal ideals in two ways, say $I=P_1\...
3
votes
0answers
116 views

Questions about the maximal irreducible components of a space

Suppose $A$ is a commutative ring with an identity, $X$ denotes its prime spectrum, that is, $X=spec(A)$, then there is a conclusion says that the maximal irreducible components of $X$ are the closed ...
4
votes
1answer
111 views

Proof that ideal of Plücker relations is a prime ideal

I am reading section 8.4 of Fulton's Young tableaux where he defines a certain ring as follows. Fix a complex vector space $E$ of dimension $m$ and integers $d_1,\ldots d_s$ such that $m \geq d_1 > ...
1
vote
1answer
62 views

Homomorphisms of a field into its valuation ring

Let $R$ be a discrete valuation ring with quotient field $K$. Let $k$ be a field contained in $R$. What are the $k$-algebra homomorphisms $\operatorname{Hom}_k(K, R)$? Are they all trivial?
1
vote
1answer
200 views

Divisor on curves, Proposition (II.6.9) from Hartshorne

I have some question related to the proof of Proposition (II.6.9) from Hartshorne's book: Let $f:X \rightarrow Y$ be a finite morphism of nonsingular curves over a field $k$. Then for any divisor ...
2
votes
0answers
82 views

If $R$ is a regular local ring module finite over $k[[x,y]]$ is $x-y^2$ irreducible in $R$?

This user deleted the following question which I think deserves to be here: Let $R$ be a regular local ring module finite over $k[[x,y]]$. Does $x-y^2$ remain irreducible over $R$? (We may assume ...
17
votes
1answer
263 views

Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the ...
7
votes
1answer
338 views

Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
6
votes
2answers
176 views

If $I\subseteq J\subseteq A$ have same image in localization by all maximal ideals, then $I=J$

I will state my question first: Suppose $I\subseteq J\subseteq A$ are two ideals in a commutative ring $A$. Furthermore, assume that for every maximal ideal $\mathfrak{m}$ of $A$, the image of $...
7
votes
3answers
475 views

The structure of a Noetherian ring in which every element is an idempotent.

Let $A$ be a ring which may not have a unity. Suppose every element $a$ of $A$ is an idempotent. i.e. $a^2 = a$. It is easily proved that $A$ is commutative. Suppose every ideal of $A$ is finitely ...
6
votes
1answer
203 views

How to show that every prime ideal is a maximal ideal if for all $a \in R$ there exists $b \in R$ such that $a^2b=a$.

Here is the full statement of the question (I thought it was a bit too long for the title). Given a commutative ring $R$ with $1 \neq 0$ such that for all $a \in R$ there exists $b \in R$ such ...
2
votes
1answer
52 views

a problem involving a homogeneous ideal and an infinite field (Matsumura, CRT, 13.1)

I am trying to solve the following problem (this is 13.1 from Matsumura's Commutative Ring Theory): Prove the following: (i) Let $R= \bigoplus_{n\ge0}R_n$ be a graded ring. Then for any $u \in R_0^*$ ...
5
votes
2answers
513 views

injective endomorphisms of finite modules need not be surjective

In the case of finite-dimensional vector spaces, an endomorphism is injective if and only if it is surjective. In the case of finitely generated modules over a commutative ring, if an endomorphism is ...
0
votes
0answers
117 views

Codimension of ideals in polynomial rings over PIDs

Let $R$ be a (commutative) principal ideal domain and let $J$ be an ideal in $R[x_1,\dots,x_n]$. Is it possible to make a general statement about the codimension (=rank/height) of $J$? As $J$ does ...
1
vote
2answers
122 views

Simple Combinatorics in finite rings

Let $g = [g_{1} g_{2} \dots g_{r}] \in \Bbb Z_{q}^{*r}$ be a given vector with each $g_{i} \in \Bbb Z_{q}^{*}$ where $\Bbb Z_{q}^{*}$ is $\Bbb Z_{q} \backslash \{0\}$ and $q > 6$ is odd. How many ...
5
votes
1answer
124 views

Images in a short exact sequence

Suppose $$ 0\to V\to W\to X\to 0\\ \downarrow\quad\quad\downarrow\quad\quad\downarrow\\ 0\to V'\to W'\to X'\to 0\\ $$ is a commutative diagram of vector spaces, with the top and bottom rows short ...
1
vote
1answer
84 views

Prüfer domains are Arithmetical rings

Suppose $R$ be a Prüfer domain. How should I Prove that it is an arithmetical ring?
0
votes
1answer
142 views

When is the formal power series ring a valuation ring?

If $K$ is a field, the formal power series ring in $1$ variable $K[[X]]$ is a discrete valuation ring. What about the many variable case? Is $K[[X_1, \ldots, X_n]]$ a valuation ring? Instead if we ...
12
votes
1answer
256 views

When some polynomials in $\mathbb Z[X]$ determine a regular sequence in $\mathbb Z[X_1,\dots,X_n]$?

Let $f_1,\dots,f_n\in\mathbb Z[X]$ be non-constant polynomials (not necessarily distinct). Is it true that $f_1(X_1),\dots,f_n(X_n)$ is a regular sequence in $\mathbb Z[X_1,\dots,X_n]$? The trivial ...
3
votes
2answers
338 views

Doubt in Hartshorne's algebraic geometry book

I'm studying by myself Algebraic Geometry and I didn't understand this part in the Hartshorne's book: I know that every polynomial $f$ in $\mathfrak a$ is written as $f=g_1f_1+\ldots + g_rf_r$, ...
7
votes
1answer
276 views

Rings in which every irreducible ideal is primary

Suppose $R$ is a commutative ring with $1$. It is well-known that if $R$ is Noetherian, then every irreducible ideal is primary (Lemma 7.12 in Atiyah & Macdonald). Is the converse true? That is: ...
0
votes
2answers
121 views

$Z(T)=Z(\mathfrak a)$, $\mathfrak a$ the ideal generated by $T$

I'm studying by myself the first chapter of Hartshorne's algebraic geometry as a introduction to this subject. I don't know how to prove this claim $Z(T)=Z(\mathfrak a)$. When we interpret the ...
1
vote
1answer
57 views

deducing inequality of polynomial degrees in a purely algebraic fashion

Let $k$ be field and $R=k[\xi_1,\cdots,\xi_r]$ be an integral domain of transcendence degree $t$ over $k$. Let $l(R_n)$ be the length of the $n^{th}$ homogeneous component of $R$. Then for $n$ large ...
6
votes
1answer
155 views

Are there Infinite Quotients of Algebraic Extensions of $\mathbb{Z}$?

It is well known that $\mathbb{Z}[a_1, \dots, a_n]/(a)$ is a finite ring if each $a_i$ is an algebraic integer and $a \neq 0.$ I suppose this statement becomes wrong if we just require those $a_i$...
3
votes
1answer
682 views

Local-global properties (localization): free, projective, injective, flat, torsion-free, etc?

Let $R$ be a commutative unital ring. We say that a property $(\ast)$ of modules is local-global when the following conditions are equivalent for any $R$-module $M$: $M$ is a $(\ast)$ $R$-module; $S^...
5
votes
3answers
3k views

If $R$ is an integral domain, then $R[[x]]$ is an integral domain

While solving another problem (specifically Exercise 7.2 in Atiyah & Macdonald's Introduction to Commutative Algebra), I got stuck in the following step: If $R$ is an integral domain, how I ...
6
votes
1answer
238 views

A commutative ring in which every prime ideal is 2-generated

Suppose $R$ is a commutative ring with 1. There are some statements that tells us if prime ideals behave in certain way, then all the ideals will behave in that way. For example, If every prime ...
2
votes
1answer
58 views

how different is the notion of an “indeterminate” from that of “algebraically independent” in relation to dimension theory?

The following is a well-known theorem in commutative algebra, see e.g. Matsumura, Commutative Ring Theory, p. 117: Let $A$ be a Noetherian ring and $X_1,\cdots,X_n$ indeterminates over $A$. Then $\...
0
votes
1answer
90 views

Do we need Gröbner bases to study factor rings of polynomials?

I'm trying to understand how we can systematically study the factor rings of polynomials over a ring K. For example imagine that we're working in $R=K[x_1,...,x_n]$ and we have the ideal $I=(p_1(x_1,.....
5
votes
0answers
85 views

regular map of Noetherian rings

We say a homomorphism of Noetherian rings $\varphi:A\rightarrow B$ is regular if $\varphi$ is flat and for every prime ideal $p$ of $A$, the fiber ring $B\otimes_Ak(p)$ is geometrically regular over $...
8
votes
2answers
319 views

Reduced rings and tensor products

I assume all rings are commutative with identity. Denote $A':=A/ \sqrt{0}$ for convenience. Question is simple: For a given ring $R$ and $R$-algebras $A,B$, does this isomorphism $(A\otimes_R B)'...
6
votes
1answer
417 views

Isomorphism of formal power series factorrings over polynomials

This problem is taken from the Hartshorne's book Algebraic Geometry, Chapter 1, Section 5, Problem 14(a). Two polynomials $f(x,y)$ and $g(x,y)$ are written in the form $$f(x,y) = f_{r}(x,y) + f_{r+...
3
votes
1answer
71 views

proving isomorphism of two $k$-algebras

Let $k$ be a field. I would like to prove that $k[x,y]/(x^3-y^2) \cong k[t^2,t^3]$. Of course, intuitively, i can readily see that this must be the case. More formally, i define a homomorphism $\phi:k[...
2
votes
2answers
132 views

continuous images of Cauchy sequences in topological groups

on page 102 of Atiyah and MacDonald's "Introduction to Commutative Algebra", they state that if $G$ and $H$ are abelian topological groups and $f$ is a continuous homomorphism from $G$ to $H$, then ...
4
votes
1answer
91 views

writing the difference of two algebraic sets as an algebraic set

Let $S$ be an algebraic set of $k^n$, where $k$ is a field and let $f \in k[x] \doteq k[x_1,\cdots,x_n]$. I am interested in expressing $Y \doteq S-Z(f)$ as an algebraic set, i.e. as the zero set of ...
4
votes
1answer
210 views

Could I see that the tensor product is right-exact using its universal property and the Yoneda lemma?

I have been doing some review with the goal of trying to understand as much as I can via universal properties and category theory (already feeling comfortable with the mundane way of doing things). ...
2
votes
2answers
139 views

can we talk about zero sets of rational functions?

Usually, when talking about algebraic sets of the affine space $\mathbb{A}^n$, we refer to zero sets of collections of polynomials in $k[x_1,\cdots,x_n]$, where $k$ is the underlying field. Question:...
10
votes
1answer
466 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
5
votes
1answer
158 views

$k$-group endomorphisms of the multiplicative group scheme for $k$ a connected ring.

I wanted to verify that for a connected ring $k$ (i.e. a ring with connected spectrum, or equivalently without idempotents other than $0$ and $1$) the group of $k$-endomorphisms of the multiplicative ...
1
vote
1answer
145 views

a theorem on the dimension of finite algebras over a field (Hartshorne)

Robin Hartshorne in his Algebraic Geometry, Theorem 1.8A(b), p. 6, says that if $B$ is an integral domain which is a finitely generated $k$-algebra, $k$ a field, and $p$ a prime ideal of $B$, then $ht(...
9
votes
1answer
223 views

Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!