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

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When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
2
votes
1answer
200 views

Expressing ideals as products of prime ideals in a commutative, Noetherian ring with unity

Let $R$ be a commutative, Noetherian ring with unity. I know that the following is true: For any ideal $I\subset R$, there are prime ideals $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ such that ...
0
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1answer
38 views

How to show that $R_{\mathfrak{m}}$ is $R$?

Let $R$ be a discrete valuation ring and $\mathfrak{m}$ its unique non-zero maximal ideal. How to show that $R_{\mathfrak{m}}$ is $R$ using definition of a discrete valuation ring? I know that ...
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1answer
320 views

How to prove that an ideal is prime?

Consider the ring of polynomials $\mathbb{C}[x,y,z]$ and the following ideal: $$I=(xy^3-z^3,x^2y-z^2,x^5z-y^5).$$ I should prove that $I$ is prime, but I don't manage to do it directly. I ...
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0answers
83 views

Question about localizations of discrete valuation rings.

Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ ...
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1answer
77 views

Algebraic study of a curve

Consider the parametric curve $C=\{(u^3,u^4,u^5)\,|\,u\in K\}$, where $K$ is an algebraic closed field with characteristic $0$. I'd like to prove that its ideal $$I(C)=\{f\in K[x,y,z]\,\mbox{such ...
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0answers
37 views

Let R be a commutative ring with identity. Prove: every nilpotent is inside every prime ideal. [duplicate]

Let R be a commutative ring with identity. Prove: every nilpotent is inside every prime ideal. Don't know what to use with this problem. Any help is usefull :)
5
votes
1answer
94 views

Prove that $\Gamma_I(\frac{M}{\Gamma_I(M)})=0$

I was trying to prove this theorem (problem): Suppose that $R$ is a commutative ring with identity, $I\unlhd R$, and $M$ an $R$-module. We define: ...
8
votes
1answer
318 views

Is an ideal generated by multilinear polynomials of different degrees always radical?

Definition. A polynomial $f\in\Bbbk[x_0,\ldots,x_n]$ is called multilinear if $\deg_{x_i}(f)=1$ for each $0\le i \le n$. In other words, $f$ is linear in each variable. If $f$ is homogeneous of degree ...
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0answers
147 views

A corollary of Grothendieck’s Finiteness Theorem

Well-known Theorem: Grothendieck’s Finiteness Theorem. Assume that $R$ is a homomorphic image of a regular (commutative Noetherian) ring. Let $\mathfrak a$ be an ideal of $R$, and let $M$ be a ...
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2answers
131 views

Embedding a module into its quotient module

I've got a very basic question on tensor products. Let $R$ be a commutative integral domain, $K$ its quotient field and let $M$ be a $R$-module. Is the map $M \rightarrow K\otimes_R M$ given by ...
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1answer
145 views

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivalence does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
2
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0answers
47 views

$I(X\times Y)=I(X)+I(Y)$ for affine varieties $X$ and $Y$? [duplicate]

Let $X\subset\mathbb{A}^{n},Y\subset\mathbb{A}^{m}$ be affine varieties over $k$ algebraically closed. Then, $X\times Y\subset \mathbb{A}^{m+n}$ may be checked to be an affine variety as well; ...
3
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1answer
122 views

The square of the maximal ideal in a local ring of dimension $2$ over a field.

I found the following assertion at page 62 in Geometry of schemes by Eisenbud and Harris: Let $R$ be a local $K$-algebra of vector-space dimension $2$, where $K$ is an algebraically closed ...
7
votes
1answer
150 views

Ideal defining the nilpotent cone of $\mathfrak{gl}_n(k)$

Let $k$ be an algebraically closed field, and let $\mathfrak{g}=\mathfrak{gl}_n(k)$. Let $\mathcal{N}\subset\mathfrak{g}$ be the nilpotent cone, that is: $$\mathcal{N}=\{A\in\mathfrak{g}\mid ...
2
votes
2answers
99 views

Ideal norm in a quadratic field

Let $K=\mathbb{Q}[\sqrt{d}]$ be a quadratic field with discriminant $d_K$, let $\mathfrak{a}=(a,\frac{b-\sqrt{d_K}}{2})$ be an ideal. Does the norm $N(\mathfrak{a})=a$? How to prove it?
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votes
1answer
137 views

Free resolution of $(x^2, y^2, xy+yz)$

Let $S = k[x,y,z]$ ($k$ field) and let $I$ be the ideal $(x^2,y^2,xy+yz)$. I computed a minimal free resolution of $S/I$, and the dimensions of the free modules in the resolution are 1,3,3,1. (Just ...
2
votes
1answer
190 views

Help in a proof in Sharp's Steps in Commutative Algebra

I'm studying the Sharp's book of commutative algebra, and I need a help in this proof why $S_0$ is a subalgebra of $S$, maybe because my lack of experience of this subject, I found myself a little ...
5
votes
0answers
64 views

Classification of radical ideals of $\mathbb{Z}[X]$ [duplicate]

The prime ideals of $\mathbb{Z}[X]$ are $0$ $(p)$ for prime integer $p$ $(f)$ for irreducible polynomial $f$ $(p, f)$ for prime integer $p$ and irreducible polynomial $f$ that remains ...
4
votes
1answer
135 views

How to compute the ring of rational functions over a Noetherian ring?

In exercise I.XXI of Geometry of Schemes by Eisenbud and Harris, one is asked to compute the ring of rational functions, firstly of an integral domain, then of a general Noetherian ring $R$, ...
2
votes
0answers
97 views

The integral closure of a power series ring over a field

Let $k$ be a field of characteristic $p$ and $K$ the field of fractions of the formal power series ring $k[[X_1,\dots,X_n]]$. Let $L$ be a finite purely inseparable field extension of $K$, then there ...
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0answers
54 views

Weakening the assumption that an ideal is maximal

This question is from ChI, $\S{3}$ of Serge Lang's Algebraic Number Theory. Let $A$ be a commutative integral domain, integrally closed in its quotient field $K$, and let $E$ be a finite extension of ...
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votes
2answers
109 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 ...
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votes
2answers
629 views

Free modules over commutative rings. [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 ...
2
votes
0answers
26 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
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0answers
63 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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0answers
140 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 $p \in P$, $p$ ...
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1answer
93 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 ...
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1answer
128 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
94 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
132 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 ...
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0answers
93 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 ...
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votes
1answer
88 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 > ...
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1answer
55 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?
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1answer
161 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 ...
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0answers
78 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 ...
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1answer
191 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 ...
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1answer
182 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 ...
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votes
2answers
106 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 ...
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3answers
359 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 ...
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1answer
159 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 ...
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1answer
41 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 $u \in R_0^*$ ...
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2answers
219 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 ...
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0answers
85 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 ...
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vote
2answers
109 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 ...
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1answer
110 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 ...
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1answer
74 views

Prüfer domains are Arithmetical rings

Suppose $R$ be a Prüfer domain. How should I Prove that it is an arithmetical ring?
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1answer
105 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 ...
11
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1answer
224 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 ...
3
votes
2answers
248 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 $f=g_1f_1+\ldots + g_rf_r$, ...