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

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6
votes
1answer
384 views

Length of maximal chain of prime ideals equals transcendence degree of fraction field?

I've been reading some commutative algebra, but have been struggling with this idea for a while. Let $k$ be a field, and let $A=k[x_1,\dots,x_n]$ be a finitely generated integral domain, such that ...
10
votes
1answer
275 views

Geometrical interpretation of $I(X_1\cap X_2)\neq I(X_1)+I(X_2)$, $X_i$ algebraic sets in $\mathbb{A}^n$

Edit: I should point out that I'm working over an algebraically closed field $k$. Let $X_1,X_2\subset\mathbb{A}^n$ be affine algebraic sets. Show that $I(X_1\cap X_2)=\sqrt{I(X_1)+I(X_2)}$. Show ...
6
votes
3answers
137 views

On a proof about locally nilpotent homomorphisms.

Let $A$ be a commutative ring. I have a short question about the small result (Proposition 2.5 of Lang's book on Algebra, pg. 418) that if $M$ is an $A$ module, and $a\in A$, then $a_M$ defined by ...
4
votes
2answers
1k views

Why is the localization of a commutative Noetherian ring still Noetherian?

This is an unproven proposition I've come across in multiple places. Suppose $A$ is a commutative Noetherian ring, and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is Noetherian. Why is this? ...
22
votes
4answers
1k views

Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module

Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module? The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't ...
4
votes
1answer
110 views

$C$ is irreducible iff $C=\mathscr{Z}(\mathfrak{p})$ for some prime ideal $\mathfrak{p}$?

Let $A$ be a commutative Noetherian ring, and $C$ a closed subset of $\operatorname{Spec}(A)$. In some reading, it is an unproven proposition that $C$ is irreducible iff ...
5
votes
1answer
225 views

Finite injective dimension of the residue field implies that the ring is regular

Let $(R,\mathfrak m,k)$ be a noetherian local ring. If $\operatorname{inj dim}_R k$ is finite, then $R$ is regular. This is exercise 3.1.26 from Bruns and Herzog, Cohen-Macaulay Rings. I don't ...
8
votes
1answer
536 views

Construct ideals in $\mathbb Z[x]$ with a given least number of generators

How do you construct, for each $n\geq 1$, an ideal in $\mathbb Z[x]$ of the form $(a_1,a_2,\dots,a_n)$ with $a_i\in \mathbb Z[x]$ such that it is impossible to have ...
3
votes
3answers
851 views

Finitely generated ideals in a Boolean ring are principal, why?

The classical book on commutative algebra Introduction to Commutative Algebra, by Atiyah and Macdonald, has the following as exercise I.11. A ring is Boolean if $x^2=x$ for any $x$ of $A$. In a ...
2
votes
0answers
235 views

does it make sense to tensor a simplicial set with a simplicial ring?

Given a simplicial commutative ring $A$ and a simplicial set $K$ does $K \otimes A$ make sense as a (commutative) simplicial ring? I'm asking as I've seen the expression $S^n \otimes A$ written down ...
1
vote
1answer
373 views

Is there a Noetherian faithful module over a non-noetherian ring?

Let $R$ be a ring which is not Noetherian. Let $M$ be an $R$-module whose annihilator is trivial. Is it possible for $M$ to be Noetherian? Intuitively, it seems like the answer should be no, as ...
6
votes
2answers
2k views

When is a product of two ideals strictly included in their intersection?

Let $I,J$ two ideals in a ring $R$. The product of ideals $IJ$ is included in $I \cap J$. For example we have equality in $\mathbb{Z}$ if generators have no common nontrival factors, in a ring $R$ ...
7
votes
1answer
273 views

Faithfully Flat Ring Homomorphism of Power Series

Let $R$ be a one-dimensional local ring and let $f:R[[x]][y] \rightarrow R[y][[x]]$ be the inclusion map. How can I show that $f$ is a faithfully flat ring homomorphism? Or can you give me a ...
2
votes
4answers
591 views

Prime ideals in polynomials rings

Let $A$ be a commutative ring, $\mathbb{q}\subset A$ an ideal of $A$, and $\mathbb{q}A[x]$ the ideal of $A[x]$ generated by $\mathbb{q}$ (consists of the polynomials with coefficients in ...
11
votes
3answers
519 views

Direct way to show: $\operatorname{Spec}(A)$ is $T_1$ $\Rightarrow$ $\operatorname{Spec}(A)$ is Hausdorff

In the book of Atiyah and MacDonald, I was doing exercise 3.11. One has to show that for a ring $A$, the following are equivalent: $A/\mathfrak{N}$ is absolute flat, where $\mathfrak{N}$ is the ...
1
vote
0answers
54 views

Is there a injective $A$-map from $A^m$ to $A^n$ when $m>n$ [duplicate]

Possible Duplicate: $A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$ Let $A$ be a commutative ring with identity, $m>n$ be positive integers. Is there an injective ...
7
votes
3answers
223 views

$\operatorname{Spec} (A)$ as a topological space satisfying the $T_0$ axiom

I have been spending a few days now proving the last bit of the following problem of Atiyah Macdonald: Prove that $X = \operatorname{Spec}(A)$ as a topological space with the Zariski Topology ...
6
votes
2answers
350 views

Intersection of algebraic sets not equal to $\{0\}$?

I was hoping to ask a small follow up to the question I asked here. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll take the definition $\dim\ V=\deg_k(k(x))$, ...
4
votes
1answer
283 views

Dimension of its irreducible components in Elimination Theory.

There is a small result I don't understand. To preface, for an algebraic variety $V\subset\mathbb{A}^n$ over some field $F$, one defines $\dim V=\operatorname{trdeg}(F(x)/F)$ for a generic point ...
9
votes
0answers
348 views

Trivial intersection of algebraic sets?

The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
1
vote
1answer
73 views

How is this lower bound of the dimension of a homogeneous algebraic variety reached?

I'm hoping to see how the following bound is reached. For an algebraic variety $V\subset\mathbb{A}^n$ over some field $F$, one defines $\dim V=\operatorname{trdeg}(F(x)/F)$ for a generic point $(x)$ ...
1
vote
1answer
206 views

fractional ideals

If $D$ is a domain and $K$ is field then for $x\in K$, $xD$ is a fractional ideals of $D$. If $xD$ and $yD$ are two fractional ideals then is true or not $xyD\subseteq xD$ ?. Thanks
3
votes
2answers
528 views

Inverse of Elements Modulo the Maximal Ideals of the Ring of Continuous functions on $[0,1]$

Let $R$ be the ring of continuous functions on $[0,1]$ under pointwise addition and multiplication, $c \in [0,1]$, and $M_c$ the ideal defined by the set of $f\in R$ that vanish at $c$. It is ...
4
votes
1answer
127 views

Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\backslash 0$?

Let $k$ be a field. Given $f, g \in k[x,y]$ coprime, why can we find $u,v \in k[x,y]$ such that $uf + vg \in k[x]\backslash 0$? I can do it for specific polynomials, but I'm struggling to structure a ...
5
votes
1answer
432 views

Structure Sheaf of the Spectrum of a Ring

Let $A$ be a ring and $X$ be the spectrum of $A$ with the Zariski topology. For an element $f\in A$ let $X_f:=\{p\subset A\text{ prime ideal }\,|\,f\notin p\}$; the $X_f$ form a basis of the topology ...
5
votes
2answers
1k views

Finding generators of an ideal / showing an ideal is prime

Suppose I have an ideal like $\mathrm{rad}(\langle y - x^2, z - x^3 \rangle)$. How do I go about finding generators for the ideal? If I could show that $\langle y - x^2, z - x^3 \rangle$ is a ...
2
votes
1answer
489 views

Prime ideals in a polynomial ring

Let $R$ be a Noetherian ring, and let $\mathfrak p$ be a prime of $R$ of codimension $d$. Suppose that $P\subset R[X]$, $P$ prime, intersects $R$ in $\mathfrak p$. Prove that if $P\neq\mathfrak ...
11
votes
1answer
899 views

How to compute localizations of quotients of polynomial rings

At the moment I'm trying to understand the concept of localizations of rings / modules. I have done some exercises (using the book of Atiyah / MacDonald) and I will do some more, but a more practical ...
1
vote
1answer
77 views

How to show a given algebra is not generated by one element

Suppose I have a $k$-algebra $k[x,y]/\langle f\rangle$, where $f$ is a (given, fixed) irreducible polynomial. What are the strategies for showing that this isn't generated by one element as a ...
3
votes
1answer
146 views

Hilbert series and initial ideals

Suppose $S$ is a polynomial ring, $I$ an ideal and $<$ some term order. Why is the Hilbert series of $S/I$ the same as the Hilbert series of $S/\mathrm{in}_<(I)$? I truly suspect the answer ...
4
votes
4answers
190 views

How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?

An ideal $M$ of a commutative ring $A$ (with unity) is maximal iff $A/M$ is a field. This is easy with the correspondence of ideals of $A/I$ with ideals of $A$ containing $I$, but how can you prove ...
3
votes
1answer
165 views

subalgebras of a polynomial ring

If $k$ is a field, I know that any subalgebra $A \subset k[x]$ is finitely generated, but I wonder if there is a good algorithm to find a set of generators for $A$. In particular, if $(f) \subset ...
7
votes
1answer
263 views

(Possibly) alternative statement of Hilbert's Nullstellensatz

In my notes, there is a statement entitled "Nulstellensatz version 2": If $k = \bar{k}$, and $\mathfrak{m} \subseteq k[x_1,\ldots,x_n]$ is a maximal ideal, then $k[x_1,\ldots,x_n]/\mathfrak{m} ...
6
votes
1answer
230 views

Equivalent condition for flatness of an A-module (Atiyah-MacDonald ex. 2.26)

I would like to solve the following exercise (2.26) from Atiyah & MacDonald's "Introduction to Commutative Algebra": If $M$ is an $A$-module (where $A$ is a commutative ring), then: $$M \text{ is ...
2
votes
1answer
91 views

Interpreting an $S^{-1}R$-module as an $R$-module.

Can one do it? I'm trying to prove that $S^{-1}I$ is an injective $S^{-1}R$-module whenever $I$ is an injective $R$-module. So I need to start with a situation where I have: (i) $S^{-1}R$-modules ...
11
votes
2answers
1k views

$A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$

I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold. This is exercise 2.11 ...
4
votes
2answers
280 views

$A$ an absolutely flat ring $\Rightarrow$ $S^{-1}A$ is absolutely flat

I was doing some exercises in the book of Atiyah / MacDonald on Commutative Algebra, and I'm a little "stuck" with number 3.10 (i): If $A$ is an absolutely flat ring and $S\subseteq A$ a ...
17
votes
2answers
705 views

Is the radical of an irreducible ideal irreducible?

Fix a commutative ring $R$. Recall that an ideal $I$ of $R$ is irreducible if $I = J_1 \cap J_2$ for ideals $J_1$ and $J_2$ only when either $I = J_1$ or $I = J_2$. Question : Assume that $I$ is an ...
4
votes
1answer
1k views

An ideal is homogeneous if and only if it can be generated by homogeneous elements

Let $S$ be a graded ring with decomposition $S = \bigoplus_{d \geq 1} S_d$, where the $S_d$ are additive abelian groups such that $S_d S_e \subseteq S_{d+e}$ for $e,d \geq 1$. An element in $S_d$ is ...
9
votes
2answers
586 views

Rings whose spectrum is Hausdorff

Let $A$ be a commutative ring with $1$ and consider the Zariski topology on $\operatorname{Spec}(A)$. When will $\operatorname{Spec}(A)$ be a Hausdorff space? If $A$ has positive or infinite ...
2
votes
1answer
67 views

What is the length of the following local ring

Let $f:Y\to X$ be a finite etale cover of smooth projective connected varieties. (Or, just a finite degree connected topological cover of connected Riemann surfaces.) Let $y\in Y$ and let $x=f(y)$. ...
9
votes
1answer
119 views

Curious about Hilbert-Zariski theorem involving homogeneous variety and set of zeroes.

I got myself in a confusing situation the other week while trying to read a bit of algebraic geometry. I'm hoping someone can pull me out. Suppose $k$ is a field, and $V$ a homogeneous variety with ...
2
votes
1answer
144 views

how does one intersect the diagonal with a graph on the surface $X\times X$

I want to do a concrete example of an intersection product for myself. Consider the endomorphism $f:\mathbf{P}^1_k\to \mathbf{P}^1_k$ given by $(x:y)\to (y:x)$. It has precisely two fixed points: ...
12
votes
2answers
354 views

Integral domain with fraction field equal to $\mathbb{R}$

I wonder if there is an integral domain $A\subseteq \mathbb{R}$ which is not a field, and such that the field of fractions of $A$ is equal to $\mathbb{R}$? Edit: here as a possible direction: it is ...
8
votes
1answer
145 views

Generators of a certain ideal

Crossposted on MathOverflow. The MathOverflow version of the question has been rewritten. For the sake of completeness, I pasted it here in a condensed form. I also deleted the old version. Let $K$ ...
5
votes
1answer
125 views

Three maximal ideals lying over $3\mathbb{Z}$?

A few weeks ago I asked a question about finding the number of maximal ideals lying above $3\mathbb{Z}$ in $B$, where $B$ is the integral closure of $\mathbb{Z}$ in a splitting extension ...
4
votes
1answer
114 views

Mod-$R$, Mod-$S$ and Mod-$R \otimes S$

Let $R,S,T$ be commutative rings and assume that $R,S$ are $T$-algebras. In an answer to this question, Pierre-Yves Gaillard gives an example of an $R \otimes_T S$-module that cannot be written as ...
16
votes
2answers
902 views

Compactness of $\operatorname{Spec}(A)$

In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact. Now ...
6
votes
1answer
220 views

Modules over a tensor product

Let $k$ be a field. Suppose $A$ and $B$ are two commutative $k$-algebras. Let $M$ be a finite $A\otimes_k B$-module. Can one find a finite $A$-module $N$ and a finite $B$-module $L$ such that $M ...
5
votes
1answer
97 views

Does the regularity of $A$ imply the regularity of $A[X]$?

Let $A$ be a commutative Noetherian ring. We say it is regular if its localization at every prime ideal is a regular local ring. If this is the case, is it true that $A[X]$ is regular?