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

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3
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2answers
853 views

Principal prime ideals are minimal among prime ideals in a UFD

Fulton, "Algebraic Curves," Exercise 1.39(a): Let $R$ be a UFD, and $P = (t)$ a principal, proper, prime ideal. Show there is no prime ideal $Q$ with $0 \subset Q \subset P$. After being ...
5
votes
0answers
337 views

radical of an ideal

Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. We define $L$-radical of $I$, denoted by $\sqrt[L]{I}$, the intersection of all primary ideals of $R$ containing $I$. It is ...
3
votes
2answers
1k views

UFDs are integrally closed

Let $A$ be a UFD, $K$ its field of fractions, and $f$ an element of $A[T]$ a monic polynomial. I'm trying to prove that if $f$ has a root $\alpha \in K$, then in fact $\alpha \in A$. I'm trying to ...
1
vote
1answer
106 views

Alternate proof of a corollary about integral dependence of rings

I am working on an alternative proof of Corollary 5.9, p.61 in Atiyah - MacDonald, "Introduction to Commutative Algebra". The Corollary reads as follows: "If $A \subseteq B$ are rings, $B$ is ...
1
vote
1answer
138 views

Simplifying quotient or localisation of a polynomial ring

Let $R$ be a commutative unital ring and $g\in R[X]$ a polynomial with the property that $g(0)$ is a unit in $R$ and $g(1)=1$. Is there any possible way to understand either $$R[X]/g$$ or $$g^{-1}R[X]...
1
vote
1answer
207 views

An example of a primeless (i.e. module without prime submodule) and projective module

Please, give an example of a module $M$ such that $M$ is primeless (i.e. without prime submodule) and projective. Thanks for your attention.
8
votes
2answers
821 views

A counterexample to the going down theorem

I will appreciate any enlightenment on the following which must be an exercise in a certain textbook. (I don't recognize where it comes from.) I understand that the going down property does not hold ...
5
votes
1answer
210 views

Isomorphisms involving localisation of graded rings

I have been trying to establish an isomorphic concerning graded rings, and there is a last step that I'm confused about. Let $R$ be a $\Bbb{Z}$ - graded ring. Let $f$ be a homogeneous non-nilpotent ...
3
votes
1answer
57 views

Free module, $\mathbb{Z}[a]$ over $\mathbb{Z}[(a+1)^2]$ for transcendental number a

I'm trying to prove that for a transcendental number $a$ the module $\mathbb{Z}[a]$ over $\mathbb{Z}[(a+1)^2]$ is free. For $\mathbb{Z}[a+1]$ over $\mathbb{Z}[(a+1)^2]$, the basis is $\{1,a+1\}$. What ...
4
votes
1answer
365 views

Calculating an example of the tensor product

I would like to show that $\mathbb{Z}/8 \otimes_{\mathbb{Z}} \mathbb{Z}_{\langle 2 \rangle} \cong \mathbb{Z} / 8$. If we let $S = \mathbb{Z} \setminus \langle 2 \rangle$, then $$\mathbb{Z}/8 \...
3
votes
1answer
631 views

The going-up theorem

I am reading Introduction to Commutative Algebra / Atiyah & Macdonald, Theorem 5.11 ("Going-up theorem"). The statement is: Let $A \subset B$ be rings, $B$ integral over $A$; let $p_1 \...
1
vote
1answer
425 views

Finding the tensor product of two finitely generated $R$-modules, where $R$ is a PID

I was asked to write down what the tensor product of two finitely generated $R$-modules $M,N$ is over a commutative ring $R$, which is a PID. I know that if $f \in R$, then $M \otimes_R R /\langle f \...
3
votes
2answers
228 views

Showing that if $R$ is local and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$.

Let $R$ be a local ring, and let $\mathfrak m$ be the maximal ideal of $R$. Let $M$ be an $R$-module. I understand that $M \otimes_R (R / \mathfrak m)$ is isomorphic to $M / \mathfrak m M$, but I ...
2
votes
2answers
261 views

Factoring a ring homomorphism

From Atiyah-Macdonald, bottom of page 9: "Let $f: A \to B$ be a ring homomorphism. ... We can factorize $f$ as follows: $$ A \xrightarrow{p} f(A) \xrightarrow{j} B$$ where $p$ is surjective and $j$...
1
vote
1answer
175 views

localization of rings and polynomial functions

Let $f$ and $g$ be two polynomials (polynomial functions in $n$ variables); if in some localization of the ring $k[X_1,\ldots, X_n]$ exists the class $\frac{f}{g}$, it defines in a unique way the ...
1
vote
1answer
87 views

Weighted initial ideal versus lex or graded reverse lex initial ideal

By imposing certain weights $\mathbf{w}$ on the variables, say, of a polynomial ring $k[x_1,\ldots, x_n]$, I read that we may obtain the initial ideal $in_{\mathbf{w}}(I)$ of an ideal $I$ with respect ...
4
votes
2answers
309 views

Finite extension of residue fields of DVR's

Let $R$ be a DVR with $K = Quot(R)$ and residue field $k$. Let $k'/k$ be a finite field extension. I would like to have a reference for the following statement (or to see, that it is not true): There ...
6
votes
4answers
749 views

Tensor product of $\mathbb R$ and $\mathbb C$ over $\mathbb R$.

$$ \mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} = \;? $$ I guess this guy is just $\mathbb{C}$, is this correct?
4
votes
1answer
388 views

Computing the local ring of an affine variety

Let $W=V(y^{2}-x^{3}) \subseteq \mathbb{A}^{2}$ and $k$ algebraically closed. Clearly the dimension of the tangent space at the origin is $2$. I want to compute this using the definition the fact that ...
13
votes
2answers
1k views

Fields finitely generated as $\mathbb Z$-algebras are finite?

Suppose $k$ is a field that is finitely generated as a ${\mathbb Z}$-algebra. (That is, $k$ is a quotient of ${\mathbb Z}[X_1,\dots,X_n]$ for some $n$). Does it follow that $k$ is finite?
1
vote
0answers
187 views

Radical of annihilator of a module

I met the following problem when I studied graded ring theory. I have no idea to solve it. Please help me. Thank you very much ! Let $R$ be a commutative $\mathbb{Z}$-graded ring, $M$ is a graded R-...
3
votes
0answers
200 views

A yet another theorem on the different ideal of algebraic number fields

I think I came up with a proof of the following theorem using non-archimedian completions. But I'm not 100% sure. Is this correct? Theorem Let $A$ be a Dedekind domain, $K$ its field of fractions. ...
2
votes
1answer
116 views

Transcendental element over a field

The following is a lemma in a note of graded ring, however, I do not know how to prove it. Please help me. Thanks. Let $R$ be a commutative reduced graded ring where $R_{0}$ is a field and let $u\in ...
3
votes
1answer
127 views

Isomorphism of initial ideals and quotient rings

I have a question related to this post. Suppose $I$ and $J$ are ideals in $R=k[x_1,\ldots,x_n]$ with $In(I)\cong In(J)$ where $In(I)$ is the ideal generated by the leading term of all those $f\in I$, ...
2
votes
1answer
341 views

Bruns-Herzog problem 3.1.25

This is problem 3.1.25 (page 97) in Cohen-Macaulay Rings by Bruns and Herzog. The direction I am interested in is the following. Let $R$ be a Gorenstein local ring and $M$ a finite $R$-module. If ...
3
votes
2answers
135 views

Showing the Dimension of a Ring

Let $f$ be in $\mathbb{Z}[x,y]$ and consider the the quotient ring $\mathbb{Z}[x,y] / \langle f \rangle$. The ring $\mathbb{Z}[x,y]$ has dimension 3, and the codimension of $\langle f \rangle$ is $\le ...
3
votes
1answer
328 views

Coheight of an ideal generated by $n$ elements

Let $R$ be a noetherian ring, and let $I$ be a proper ideal in $R$. If $I$ is generated by $n$ elements, we have by Krull's Principal Ideal Theorem that the height of $I$ is at most $n$. Is it true ...
7
votes
2answers
3k views

Is every Noetherian module finitely generated?

I was just wondering whether the following statement is correct. Let R be a ring and M a noetherian R module. Then M is finitely generated.
4
votes
3answers
380 views

What is the support of a localised module?

Let $R$ be a noetherian commutative ring, and let $\mathfrak{m}$ be a maximal ideal of $R$. Let $M$ be a finitely-generated torsion $R_\mathfrak{m}$-module, considered as an $R$-module. Is it possible ...
8
votes
1answer
295 views

If $R$ is commutative, and $J\lhd I\lhd R,$ does it follow that $J\lhd R?$

$\lhd$ will stand for "is an ideal of" in this post. Let $R$ be a commutative ring, $J\lhd I\lhd R$. Does it follow that $J\lhd R?$ I don't think it does, but I'm having difficulty finding a ...
6
votes
2answers
291 views

Detecting whether something is a Dedekind domain

Consider the three rings $\mathbb{C}[x,y] / \langle x^4 + xy -1\rangle$, $\mathbb{Z}[x,y] / \langle x^4 + xy -1\rangle$ and $\mathbb{F}_2[x,y] /\langle x^4- y^3 \rangle$. I am supposed to detect ...
5
votes
2answers
828 views

How to show a ring is normal or not, and how to show the normalisation of the ring

I am confused about how to show whether a ring is normal or not. For example, consider the $k$-algebra $k[x,y] /\langle x^2 - y^3 \rangle$, which is a domain. How do I show it is not normal? Are ...
8
votes
2answers
337 views

Computing an example of Ext

Let $k$ be a field. I want to compute $\operatorname{Ext}_{k[x] / \langle x^2 \rangle}(k,k)$. However I have no idea how to do this? I cannot even think how to construct a projective resolution ...
10
votes
2answers
2k views

Finitely generated projective module

Would anyone can help me how to show that a finitely generated projective module over a local ring and PID are free? What I know about a finitely generated projective module $M$ over a PID $R$ is ...
6
votes
2answers
224 views

Question on Noetherian/Artinian properties of a graded ring

Let $R$ be a non-negatively graded Noetherian ring such that $R_{0}$ is Artinian and $R_{+}$ is a nilpotent ideal. Prove that $R$ is Artinian. Give an example to show that this is false if the ...
4
votes
1answer
224 views

Ascending chain conditions on homogeneous ideals

Here is one exercise from some notes on graded rings. I tried but I got no idea to solve it. Please help me. Thanks. Let $R$ be a graded ring. Prove that $R$ is Noetherian (Artinian) if and only ...
1
vote
1answer
138 views

Prime Ideals in a Localised Ring

Let $R$ be a commutative ring, and let $S \subseteq R$ be a multiplicatively closed subset (not containing $0$). Then we construct the localised ring $R [ S^{-1} ]$. I understand that prime ideals in $...
6
votes
1answer
220 views

Does the contraction from the localized ring preserve colon ideals and ideal sums/products?

Let $A$ be a commutative ring and $B = S^{-1}A$ be its localization with respect to a certain multiplicative subset of $A$. Consider the contraction (in $A$) of colon ideals and ideal sums and ideal ...
11
votes
1answer
2k views

Definition of a finitely generated $k$ - algebra

In Miles Reid's Undergraduate Commutative Algebra he defines a ring $B$ to be finite as an $A$ - algebra if it is finite as an $A$ - module. Now what I don't understand is suppose we look at the ...
3
votes
1answer
355 views

A homomorphism between finite free modules over a local ring

The following lemma is stated in a book without a proof. How can this be proved? Lemma Let $A$ be a local ring. Let $k$ be the residue field of $A$. Let $E$ and $F$ be finite free modules over $A$. ...
-1
votes
1answer
130 views

$\hat{M_P}$ $\cong$ $\prod_{i}\hat{M_{Q_i}}$

I think I came up with the following result. But I'm not 100% sure. Is this correct? If yes, how does one prove this? Theorem? Let $A$ be a discrete valuation ring, $K$ its field of fractions. Let $L$...
2
votes
1answer
882 views

Homogenous localization and usual localization in graded rings

Let $R$ be a graded ring. There are two ways to take the localization of $R$. Let $\mathfrak{p}$ be a homogenous prime ideal, $T$ be the set of all homogenous elements of $R\setminus \mathfrak{p}$. ...
1
vote
1answer
1k views

A maximal ideal among those avoiding a multiplicative set is prime

Let $S$ be a multiplicatively closed subset of a ring $R$, and let $I$ be an ideal of $R$ which is maximal among ideals disjoint from $S$. Show that $I$ is prime. If $R$ is an integral domain, explain ...
4
votes
2answers
169 views

Confusion about local properties + trying to show a finite morphism is quasi-finite

Let $A$ be a commutative ring and let $B$ be a finite $A$-algebra. Let $f:A \to B$ be a ring homomorphism. I want to show that whenever $\mathfrak{p} \subseteq A$ is a prime ideal, then there are ...
1
vote
1answer
473 views

Finding all simple $R$ modules of a ring.

I was hoping someone had an idea on how to go about solving the following; Find (up to isomorphism) all simple R-modules where i) $R = \begin{pmatrix} \mathbb{Z}/15 \mathbb{Z} & \mathbb{Z}/15 \...
1
vote
1answer
209 views

Notation question for denoting an ideal of a polynomial ring

Let $k$ be a field. Let $q=(x,y^2)$ be an ideal of $k[x,y]$. What exactly does the notation $q=(x,y^2)$ mean, i.e. what kind of elements does $q$ contain? Is it the set of all elements $\alpha x + \...
14
votes
5answers
2k views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
6
votes
3answers
1k views

An ideal that is maximal among non-finitely generated ideals is prime.

I've been doing some old exam problems and I've come across a problem that I've answered, but my gut is telling me that there's something I'm glossing over. Let $R$ be a commutative ring with ...
9
votes
3answers
2k views

If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field. [duplicate]

Possible Duplicate: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I need to prove this result, but the only starting point I think of is to ...
2
votes
2answers
268 views

On the sum of all the simple submodules of a module

$R$ ring and $M$ a left $R$-module. Call $\mathrm{Soc}\;M$ the sum of all the simple submodules of $M$. Then $M$ is artinian if and only if $\lambda_R(\mathrm{Soc}(M))<\infty$ and for very $0\neq ...