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

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6
votes
1answer
224 views

Points and sheaf of functions for some schemes

I am (slowly) reading Eisenbud and Harris and trying to get my head around (affine) schemes. Let $R$ be a ring, and $X=\text{spec}(R)$, as usual with the Zariski topology. We have a basis of open ...
11
votes
2answers
548 views

Examples demonstrating that the finitely generated hypothesis in Nakayama's lemma is necessary

Recall that Nakayama's lemma states that Let $R$ be a commutative ring with unity, and let $J$ be the Jacobson radical of $R$ (the intersection of all the maximal ideals of $R$). For any finitely ...
8
votes
2answers
1k views

Is fibre product of varieties irreducible (integral)?

Let $k$ be an algebraically closed field and $X,Y$ varieties (i.e. integral, separated schemes of finite type over $k$). Is the fibre product $X \times_k Y$ necessary irreducible or integral? I ...
47
votes
5answers
1k views

How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a ...
2
votes
2answers
803 views

Wanting to show $a+x$ is a unit for unit $a$ and nilpotent $x$ [duplicate]

Possible Duplicate: Units and Nilpotents If $a$ is a unit and $x$ is nilpotent, I'm trying to show that $a+x$ is a unit. Pf.: If $a$ is a unit, there exists a non-zero invertible element ...
0
votes
2answers
160 views

Radical and nilrad

I'm trying to prove that the set $\mathrm{nilrad}(A)$ of nilpotent elements of $A$ is an ideal Pf/ if $g\in\mathrm{nilrad}(A)$, then $g^n = 0$, for some $n>0$. Let $h$ be an element of ...
13
votes
1answer
525 views

Exercise 2.17(d) of Eisenbud's Commutative Algebra

First some notation: Let $P$ be a homogeneous prime ideal of a $\Bbb{Z}$ - graded ring $R$, $U$ the multiplicative subset of all homogeneous elements not in $P$. Suppose that there exists a ...
11
votes
4answers
282 views

Show $\mathbb{Q}[x,y]/\langle x,y \rangle$ is Not Projective as a $\mathbb{Q}[x,y]$-Module.

Disclaimer: Though I have been re-reading my notes, and have scanned the relevant texts, my commutative algebra is quite rusty, so I may be overlooking something basic. I want to show $\mathbb{Q} ...
4
votes
1answer
109 views

If $g$ and $g\circ f$ are graded homomorphisms, must $f$ be graded?

Question Let $A$ be a graded ring (always commutative with identity) and $M,N$ and $P$ be graded $A$-modules. Let $f:M \longrightarrow N$ and $g:N \longrightarrow P$ be $A$-module homomorphisms ...
7
votes
1answer
192 views

The product of two spectral spaces

Notice: the following statements about the product topologies are all Cartesian product topology, we are in the category of topology not the category of schemes. In this page of sober space, it said ...
4
votes
1answer
301 views

A formula for the minimum number of generators of a module over a semilocal ring

Let $R$ be a commutative ring with only finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_r$. Let $M$ be a finitely generated $R$-module. Then $$\mu_R(M)=\max\{\dim_{R/\mathfrak ...
6
votes
2answers
242 views

How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$

Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that ...
3
votes
3answers
810 views

Does $A$ a UFD imply that $A[T]$ is also a UFD?

I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD. The only thing I am sure I could try to use is Gauss's lemma. Also, how can we deduce that the polynomial rings ...
6
votes
3answers
1k views

When the localization of a ring is a field

Let $R$ be a commutative noetherian ring with no nonzero nilpotents. Let $p$ be a minimal prime of $R$. Could you help me to prove that $R_p$ is a field?
2
votes
1answer
113 views

An equivalent condition for having finite length

Let $R$ be a commutative noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite length if and only if $M_p=0$ for every non-maximal prime ideals $p$?
7
votes
2answers
173 views

Relation between projective modules over $R$ and $R[T]$

Let $R$ be a commutative ring and $R[U]$ the polynomial ring in one variable. What is the relation between projective modules over $R$ and projective modules over $R[U]$? Is every projective module ...
2
votes
3answers
73 views

If $Ra$ is free for $a\neq 0,$ is $a$ regular?

Let $R$ be a commutative ring with unity, and $0\neq a\in R.$ We will say that an element $x\in R$ is linearly independent if $\{x\}$ is a linearly independent set. A non-zero element of $R$ is called ...
3
votes
2answers
591 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
276 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 ...
2
votes
2answers
663 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
94 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
129 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 ...
1
vote
1answer
179 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.
6
votes
3answers
619 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 recognise where it comes from.) I understand that the going down property does not hold ...
5
votes
1answer
202 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
362 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 ...
2
votes
1answer
486 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
386 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
195 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
227 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 ...
1
vote
1answer
139 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
72 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 ...
3
votes
2answers
227 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
536 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
274 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
783 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
159 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 ...
3
votes
0answers
158 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
110 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
113 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
293 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
128 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 ...
2
votes
1answer
290 views

Krull's Principal Ideal Theorem

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 ...
5
votes
2answers
1k 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.
3
votes
3answers
356 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 ...
7
votes
1answer
260 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
273 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 ...
4
votes
2answers
631 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 ...
7
votes
2answers
289 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 ...