Tagged Questions

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

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0
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1answer
25 views

every ideal that is not intersect $S$ is prime ideal?

every ideal that is not intersect $S$ where $S$ is multiplicative closed is prime ideal? I know that maximal ideals among those are prime ideals. But what about other ideals that is not intersect $S$. ...
0
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0answers
53 views

Tensoring an exact sequence of $R$-modules with $R/x$

Let $R$ be a commutative ring with an $R$-module $M$, and let $x \in R$ be an $M$-regular element. Then tensoring any short exact sequence $0 \to B \to A \to M \to 0$ with $R/x$ yields a short exact ...
1
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1answer
47 views

Does $ax\in\mathfrak{m}I$ with $x\in I\setminus\mathfrak{m}I$ and $a \in R$ imply $a\in\mathfrak{m}$ for an invertible fractional $R$-ideal $I$?

Let $R$ be an integral domain, $\mathfrak{m}$ a maximal ideal of $R$, and $I$ an invertible fractional $R$-ideal. If $x \in I \setminus \mathfrak{m}I$ and $a \not\in \mathfrak{m}$, do we have $ax ...
1
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2answers
53 views

It's true that a valuation ring $R$ in the quotient field of a normal ring $A$ contain $A$?

Let $A$ be a finitely generated $k$-algebra ($k$ algebraically closed) of dimension one, integrally closed in its quotient field $K$. Let $R\subseteq K$ be a valuation ring. It's true that $A\subseteq ...
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0answers
36 views

Prove there are some elements in a commutative module [duplicate]

let R be a commutative ring and I is an ideal in R and also M is finite generating module on R. If $\varphi:\:M \to M$ be a homomorphism and $\varphi(M)\subset IM$ .Prove there are some elements ...
2
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0answers
58 views

Non-Noetherian subring of F[X,Y]

I am trying to prove that, for a given field $F$, the subring $$R:=\{p(X,Y)=\sum c_{ij}X^iY^j \in F[X,Y] : c_{0j}=c_{j0}=0 \text{ whenever } j>0\}$$ of $F[X,Y]$ is not Noetherian. I think I ...
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2answers
67 views

How to show rational function field of an affine subvariety with dim>0 is not algebraically closed?

I do not know how to show the following statement. If $X\subset A^n$ is an irreducible subvariety, $\dim X>0$, then the rational function field of $X$, $K(X)$ is not algebraic closed. What ...
6
votes
1answer
70 views

Generalizing the Big Omega function to Integral Domains

The $\Omega(n)$ function counts the total number of prime factors of $n$ counting multiplicity. Obviously, this definition extends to any Unique Factorization Domain. I have two follow up questions: ...
1
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2answers
83 views

Prime ideals in $k[x,y]/(xy-1)$.

Let $k$ a field. Let $f$ be the ring injective homomorphism $$ f:k[x] \rightarrow k[x,y]/(xy-1)$$ obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ ...
2
votes
1answer
95 views

Elementary motivations for free resolutions

Let $M$ be a finitely generated module over a Noetherian ring $R$ which admits a finite free resolution $0 \to F_n \to \dots \to F_0 \to M \to 0$. There is no doubt that knowing such a resolution is ...
0
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1answer
47 views

On local ring homomorphisms

Suppose I have two local rings $A$ and $B$, and suppose I have $\phi : A \rightarrow B$, which is a ring isomorphism. Does it follow then that $\phi$ is a local ring homomorphism? The point of ...
2
votes
2answers
44 views

Characterization of ideals of algebra of continuous functions on a compact space.

I was reading this planetmath page on the connections between the topology on a compact Hausdorff topological space $X$ and the maximal ideals on the algebra of continuous functions $C(X)$ on $X$, ...
1
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0answers
36 views

Commutative diagram of algebras in Atiyah and Macdonald.

On page 31 of Atiyah and Macdonald, there is a commutative diagram. It essentially says that if $B$ and $C$ are $A$-algebras with ring morphisms $f:A\to B$ and $g\colon A\to C$, and $D=B\otimes_A C$ ...
0
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0answers
36 views

Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16

This question is from the Bruns-Herzog, Cohen-Macaulay Rings, Exercise 10.1.16(a). Let $x_1,..., x_n, y, z$ be elements of $R$ such that ideals $(x_1,..., x_n,y)$ and $(x_1,..., x_n,z)$ are ...
2
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1answer
56 views

$I(Y) = \{ p(x,y,z) \in k[x,y,z] \mid p (t,t^2,t^3) = 0, \forall t \in k \}$ is prime

I've been working on the following problem from Hartshorne: Let $Y\subseteq \mathbb{ A }^3 $ be the set $Y = \{(t,t^2 , t^3) \mid t \in k \}$. Show that $Y$ is an affine variety of dimension $1$. To ...
1
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1answer
77 views

Finitely generated prime ideal and annihilator

Suppose $R$ is a commutative ring, $P$ is a prime ideal of $R$, $P$ is finitely generated, and $\operatorname{Ann}(P)=0$. Show that $$\operatorname{Ann}(P/P^2)=P.$$ These are my efforts: ...
1
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1answer
42 views

vanishing of an Ext-Functor for a finite graded module of positive grade over a polynomial ring

Let $k$ be a field and $S=k[x_1,\dots,x_r]$ the polynomial ring in $r$ indeterminates. Let $M$ be a finitely-generated, graded $S$-module, such that there exists a homogeneous $M$-regular element $\xi ...
2
votes
1answer
52 views

Algebraic Curves: Valuation at a point

I would like to understand the notion of valuation on the local ring of a curve at a point. In the Book The Arithmetic of Elliptic Curves in chapter 2, Example 1.3 $$V:\ Y^{2}=X^{3}+X$$ I don't ...
1
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1answer
46 views

A relation in a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module s.t. $M=IM$. How to prove: $$\exists a \in I \text{ such that } (1-a)M=0. $$ I tried to solve: ...
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3answers
57 views

Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors. [duplicate]

Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors. I begin to assume that $A$ has no zero divisors but I don't know continue... \ How would be this proof? thanks! :)
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0answers
47 views

Rings and modules

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-modules. Now let $I$ be a proper ideal of $R$. What is the structure of $I$. Is it true that $I$ is a direct sum of cyclic ...
0
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1answer
67 views

Prove that a subset is a finitely generated subring

Consider $\mathbb{A}^2$ with $\rho : (x, y) \mapsto (-x, -y)$. Can anyone help me prove that $S = \{f \in \mathbb{C}[x, y] : f \circ \rho = f\}$ is a finitely generated subring? Also, can $S$ be ...
3
votes
1answer
76 views

Atiyah-Macdonald 5.2

Exercise 5.2 in Atiyah-Macdonald asks to show the following: "Let $A$ be a subring of a ring $B$ such that $B$ is integral over $A$, and let $f: A \to \Omega$ be a homomorphism of $A$ into an ...
1
vote
1answer
114 views

A relation involving an endomorphism of a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module. (Usually in this problem $R$ includes $1_R$.) Let $\phi : M \to M$ be an $R$-homomorphism, and ...
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votes
2answers
37 views

Prove that T is not a zerodivisor in A[T]

Let A be any ring, consider the polynomial ring A[T]. Prove that T is not a zerodivisor in A[T]. Generalise the argument to prove that a monic polynomial $$ f=T^n+a_{n-1}T^{n-1}+\dots+a_0 $$ is ...
7
votes
1answer
116 views

Tensor Product, Exterior Power and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $$0\to K\to M\otimes M\to M\wedge M\to 0$$ ...
2
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2answers
83 views

Going-up and going-down theorems: motivation

I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book. I'm wondering if anyone could give me some basic facts/examples to help me understand why ...
1
vote
1answer
49 views

Primary ideal exercise

I have an exercise about the properties of primary ideal. It's Exercise 15.17 of "Step in commutative algebra", R. Y. Sharp. Let $(A,\mathfrak{m})$ be a local ring and $I$ be a proper ideal of $A$. ...
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0answers
40 views

A query on Veronese mapping

The Veronese mapping defined as usual on some $P^n$. Then it is certainly regular. I want to prove that the inverse map to this map is also regular. I have an idea to use projections with ...
1
vote
2answers
24 views

$rad(I)=\cap_{I\subset P,~P~prime}P$

$R$ commutative ring with unity. $I$ R-ideal. Then $rad(I)=\cap_{I\subset P,~P~prime}P$. That is, the radical of $I$ is the intersection of all prime ideals containing $I$. There is a proof of this ...
1
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1answer
53 views

If $p \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
1
vote
2answers
28 views

minimal prime ideals over the union of two prime ideals

When two subvarieties intersect properly ($X_1\cap X_2$), it should end up with a new subvariety($X_3$=$X_1\cap X_2$). I do not know how to keep track of the intersection operation from the algebraic ...
1
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1answer
32 views

“Adjugate” of an endomorphism of a finite-rank free module

If $M$ is a free module of finite rank $n$ over a commutative unitary ring and $a$ is an endomorphism of $M$, consider the endomorphism $\hat a$ of $M$ defined by the identity $$ x_1\wedge ...
0
votes
1answer
44 views

A question related to the height of a proper ideal in a Noetherian ring

Let $A$ be a Noetherian ring, and $I\subset A$ a proper ideal of height $r$. Is it true that there exist $a_1,\ldots,a_r\in I$ such that $$\operatorname{ht}(a_1,\ldots,a_j)=j$$ for all $j=1,\ldots,r$ ...
3
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1answer
33 views

Basis-free and noncommutative versions of the two-polynomials-over-ring problem (McCoy theorem etc.)

There is a rather canonical bunch of exercises in commutative algebra which tend to come up time and again on math.stackexchange: recently in #948010 and #83121, formerly in #227787 and #413788, and ...
4
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1answer
196 views

Zero divisors in $A[x_1,x_2,\dots,x_r]$

I am trying to show that if $f(x_1,x_2,\ldots, x_r) \in A[x_1,x_2,\ldots, x_r]$ is a zero divisor then there exists $a$ in $A-\{0\}$ such that $af=0$ in $A[x_1,x_2,\ldots, x_r]$. What I have ...
2
votes
0answers
49 views

Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
0
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0answers
39 views

Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
0
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1answer
44 views

Question about the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$.

Given the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$, I want to compute $V(I)$, which is the intersection of all ideals containing $I$. And I also want to prove that $I$ can't be generated by ...
0
votes
1answer
60 views

Relation between ideals in Noetherian domains.

Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$ Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and ...
2
votes
2answers
96 views

Showing an ideal with maximality condition is prime.

Let $R$ be a commutative domain and suppose that $I \subseteq R$ is an ideal of $R$ maximal with respect to the property that $I^{-1} \not\subseteq R$. Show that $I$ is a prime ideal. This is ...
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0answers
30 views

K[x,y,z,w]/(xw-yz) not UFD [duplicate]

I am trying to prove its not UFD. I started by assuming x=ab in K[v] , where v=v(xw-yz) then x-ab=(xw-yz)f, for some f in K[x,y,w,z] I tried to say that deg of a, and b is less or equal 1, and ...
1
vote
1answer
38 views

Cohen structure theorem for artinian local rings

Let $(R,m)$ be an artinian local ring. Since $m^n=0$ for some $n$, it is clear that $R$ is complete with respect to $m$-adic topology. Now i want to know that how do we state the Cohen structure ...
1
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1answer
31 views

maximal algebraically independent sets in ring extensions

Let $E/K$ be a field extension. It is a well known fact that all maximal subsets $A \subset E$ consisting of algebraically independent elements over $K$ have the same cardinality (which is by ...
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0answers
40 views

Bruns-Herzog, Cohen-Macaulay Rings, Exercise 9.1.10(c)

The following question is from the book: Bruns-Herzog, Cohen-Macaulay Rings, Exercise 9.1.10(c). Let R be a ring, I a finite generated ideal and M an R-module. Let R$_{\infty}$ be a polynomial ...
1
vote
1answer
56 views

Krull dimension in finite ring extensions

Let $K$ be a field and $R=K[a_1, \dots, a_n]$ a finite ring extension. Suppose that the degree of transcendence of $R$ over $K$ is $r$. Then the Krull dimension of $R$ is at most $r$. I would like to ...
3
votes
1answer
81 views

Localization at finitely many minimal prime ideals

Let $A$ be a commutative ring with finitely many minimal prime ideals $\{p_1,\dots,p_n\}$. Let $A_{p_1,\dots,p_n}$ be the localization of $A$ away from the minimal primes, i.e. $S^{-1}A$ where $S = ...
2
votes
1answer
70 views

Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
1
vote
1answer
45 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
0
votes
1answer
76 views

Is $\mathbb Z\oplus \mathbb Q$ a flat $\mathbb Z$-module? [closed]

Can someone please explain why $\mathbb Z\oplus \mathbb Q$ is flat or not?