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

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3
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1answer
169 views

Question on schemes and function fields

Let $X=Spec(R)$ be an irreducible noetherian scheme and $\eta$ the unique minimal prime ideal of $R$. Let $U$ and $V$ be open sets in $X$ and $p$ a point with $p\in V\subseteq U\subseteq X$. If $X$ ...
7
votes
1answer
227 views

Questions on scheme morphisms

I have some questions on scheme morphisms. I ask pardon for posting them in one thread as they are most likely not worth to be distributed into several threads. Let $X=Spec R$ be a noetherian scheme. ...
17
votes
1answer
1k views

Hom is a left-exact functor

If $0 \to A \to B\to C$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $0 \to Hom_R(M,A)\to Hom_R(M,B)\to Hom_R(M,C)$ is left exact. I proved the above, and highlighted what ...
2
votes
1answer
314 views

Does every closed set of prime ideals of a noetherian ring contain a finite dense subset?

Does every closed set of prime ideals of a noetherian (commutative) ring contain a finite dense subset? EDIT 1. Here is the motivation. In books like Mumford’s red book or Eisenbud-Harris, the ...
2
votes
2answers
229 views

Faithful extension(Is the image of a polynomial map an algebraic set? )

Let $k[x_1,\ldots,x_n]$ be a polynomial ring, $k$ be a algebraically closed field. Suppose $k[T_1,\ldots,T_m]$ is finitely generated $k$-subalgebra such that for any proper ideal $I$ of ...
3
votes
1answer
116 views

What is $A^-$ in topology?

Hartshorne Algebraic Geometry Proposition 1.5 (page5) In a notherian topological space $X$, every nomempty closed subset $Y$ can be expressed as a finite union $Y=Y_1 \cup \cdots \cup Y_r$ of ...
5
votes
2answers
1k views

Atiyah-Macdonald Ex8.3: Artinian iff finite k-algebra

Atiyah Macdonald Ex8.3 Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equvialent: (1) $A$ is Artinian (2) $A$ is a finite $k$-algebra. I have a ...
9
votes
2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
3
votes
1answer
269 views

Generating a regular sequence out of two

Here is the last problem of my final exam in "Commutative algebra" which I think, no one has solved it completely, today! Let $R$ be a commutative Noetherian ring. Let $a_1,\dots,a_n$ and ...
5
votes
2answers
217 views

Does $\operatorname{Spec}(B/\mathfrak{p}^e)=\operatorname{Spec}((A/\mathfrak{p}) \otimes_A B)$?

This is from Atiyah-Macdonald Ex7.23. (It seems that the entire problem is not needed, but I will write it just in case: Let $A$ be a Notherian ring, $f:A \to B$ a ring homomorphism of finite type. ...
2
votes
1answer
248 views

What happens geometrically when the Jacobson radical is non-zero?

I'm interested in intuition about the affine schemes of rings with a non-vanishing Jacobson radical. In the ring of real-valued continuous functions on a topological space the Jacobson radical is ...
3
votes
1answer
124 views

Atiyah-Macdonald Ex7.22: Equivalent condition for $E$ to be open in a Notherian space

Atiyah-Macdonald Ex7.22 Let $X$ be a Notherian topological space and let $E$ be a subset of $X$. Show that $E$ is open in $X$ if and only if, for each irreducible closed subset $X_0$ in $X$, either ...
11
votes
2answers
782 views

Must $k$-subalgebra of $k[x]$ be finitely generated?

Suppose $k$ is a field, $A$ is a $k$-subalgebra of the polynomial ring $k[x]$. Must $A$ be a finitely generated $k$-algebra? Thanks.
6
votes
2answers
370 views

Atiyah-Macdonald Ex 7.20 Constructible sets

Let $X$ be a topological space and let $\mathscr{F}$ be the smallest collection of subsets of $X$ which contains all open subsets of $X$ and is closed with respect to the formation of finite ...
3
votes
1answer
97 views

Dimension of Overring

Let $A$ be an (Noetherian) integral domain (of dimension one), $K$ its quotient field, $B$ a subring of $K$ such that $A\subseteq B \subseteq K$. Can we determine the dimension of $B$ in general? what ...
4
votes
1answer
213 views

Koszul algebra of a ring

I'm studying on Cohen-Macaulay Rings of Bruns-Herzog. Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $H_{\bullet}(R)$ its Koszul algebra. I found on the book (page 75) that "since ...
3
votes
2answers
224 views

Auslander-Buchsbaum and Ferrand-Vasconcelos

I'm studying on "Cohen-Macaulay rings" of Bruns-Herzog, here a link: http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false At page 65 there is ...
6
votes
2answers
1k views

Minimal free resolution

I'm studying on the book "Cohen-Macaulay rings" of Bruns-Herzog (Here's a link and an image of the page in question for those unable to use Google Books.) At page 17 it talks about minimal free ...
4
votes
0answers
152 views

A question about an example on flat families from Hartshorne. In particular, is this local ring reduced?

Is the local ring $R_p$ reduced, where $p=(a,x,y,z)$ and $R=k[a,x,y,z]/I$ and $I=(a^2(x+1)-z^2,ax(x+1)-yz,xz-ay,y^2-x^2(x+1))$ ? This comes from example III.9.8.4 in Hartshorne's algebraic geometry. ...
2
votes
1answer
233 views

Help on a proof of a Theorem of Rees

I'm studying on this book http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false on page 10 there is a Rees Theorem. I'd like to know why the ...
1
vote
1answer
865 views

Finitely generated ideal

We say that an ideal $\alpha$ of $A$ is finitely generated if $\alpha =(x_1,\cdots,x_n)=\sum_{i=1}^{n} Ax_i$, i.e. finitely generated as an $A$-module. Then how we call if $\alpha$ is generated by ...
1
vote
0answers
94 views

Hermite normal form and saturation

Recall that if $M$ is a submodule of $\mathbb{Z}^n$, then the saturation of $M$ (in $\mathbb{Q}$) is defined to be $\mathbb{Z}^n \cap (\mathbb{Q}\otimes_{\mathbb{Z}} M)$. According to an article of ...
2
votes
1answer
223 views

Determine whether two rings are isomorphic

$k$ is a field, char$(k)\neq 2$. Can $k[x,y]/(y-x^2)$ be isomorphic to $k[x,y]/(x^2+y^2-1)$ by some ring homomorphism? Edit: Since jspecter gives a nice and quick proof, I will ask another ...
9
votes
1answer
431 views

Grade of a maximal prime ideal in a Noetherian UFD

Here is an another problem in Commutative Rings by Kaplansky, p. 103, no. 15. Let $R$ be a Noetherian UFD. Let $(a,b) \not= R$ where $a,b \in R.$ Prove that any maximal prime of $(a,b)$ has grade ...
2
votes
2answers
158 views

Does a 1-dimensional noetherian domain obey cancellation law?

Someone calls this "order" which puzzles me, because I can't understand it's name. I was wondering whether these rings obey cancellation law, i.e. if $\mathfrak a\mathfrak b=\mathfrak a\mathfrak c$ ...
0
votes
1answer
389 views

Codimension and Krull's principal ideal theorem

I know that in an UFD, each minimal prime will be principal. So, let $k[x_1,...,x_n]$ be a polynomial ring over a field. Further, set $S =k[x_1,..,x_n]/P$, and suppose that this is an UFD. A ...
17
votes
5answers
2k views

Showing the set of zero-divisors is a union of prime ideals

I'm working on an exercise from Atiyah and MacDonald's Commutative Algebra, and have hit a bump on Exercise 14 of Chapter 1. In a ring $A$, let $\Sigma$ be the set of all ideals in which every ...
1
vote
1answer
131 views

About universal property(?) of injective hull

Let $M$ be a module over a commutative ring with unity. If $M\subset Q$, where $Q$ is an injective module, we know that $Q$ contains an injective hull of $M$. My question is: Can $Q$ contain several ...
3
votes
2answers
643 views

The ring of power series in z with a positive radius of convergence

Let $A$ be the ring of power series in $z$ with a positive radius of convergence, where all coefficients are complex numbers. What is the name of $A$? Does it have a special name? I want to show ...
5
votes
2answers
389 views

always irreducible polynomial?

There exists irreducible polynomials in $\mathbb{Z}[x]$ (e.g. $x^4-10x^2+1$) which is reducible modulo every prime $p$.(A proof can be found in J.S. Milne's "Fields and Galois Theory", page13. Here is ...
1
vote
2answers
286 views

A characterisation of tame ramification

The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155) Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the ...
2
votes
0answers
194 views

When does the generating set of a finite rank $A$-module reduce to a basis of the $A/\mathfrak{m}$-vector space?

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and let $k = A/\mathfrak{m}$ be the residue field. Let $M$ be a finitely generated $A$-module; so $M/\mathfrak{m}M$ is a finite dimensional ...
3
votes
0answers
91 views

Azumaya algebra and its subalgebras

I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such ...
2
votes
1answer
212 views

Zero-divisors and R-sequences

Here is an exercise which I can not solve. You may find it in Commutative Rings by Irving Kaplansky, p. 103, ex. 13. Let $R$ be a commutative ring with $1$ (not necessarily Noetherian) and $A$ an ...
4
votes
1answer
274 views

Failures of Nakayama's lemma or Krull theorem

In this post, if $I$ is a finitely generated ideal of a commutative ring $A$ with 1 such that $I^2=I$, then we know $I$ is principal. To answer this question without using Nakayama lemma, I have a ...
4
votes
1answer
188 views

Can I assume $R$ is a local ring?

I should prove this statement: Let $R$ be a ring, $M$ be a $R$-module and $P$ a projective $R$-module of finite type. If $x=\sum_i m_i\otimes p_i$ is an element in $M\otimes_R P$ such that $\sum_i ...
5
votes
1answer
198 views

transitivity of finitely generated condition

Let $A \subseteq B \subseteq C$ be rings. I know that if $B$ is a finitely generated $A$-module and $C$ is a finitely generated $B$-module, then $C$ is a finitely generated $A$-module. (Proof is in ...
1
vote
1answer
408 views

Concept of Free module in Polynomial ring

I'm studying Atiyah's commutative algebra. I have a question with free modules and the kind of thing in polynomial ring. I wrote the following so it cannot be true facts. A free $A$-module is ...
1
vote
1answer
147 views

$Spec(R)$ Noetherian and going up theorem

Let $S \subseteq R$ be commutative rings with 1 and suppose $Spec(R)$ is a Noetherian topological space. How do we show that the number of each $T \in Spec(R)$ lying over $P \in Spec(S)$ is finite? I ...
2
votes
0answers
151 views

Twisted forms of a free module

Let $R$ be a ring an let $A$ and $B$ be two $R$-algebras. If $S$ is faithfully flat over $R$ then we say that $A$ is a $S$-twisted form of $B$ if $A\otimes_R S$ and $B\otimes_R S$ are isomorphic as ...
5
votes
2answers
820 views

I want a proof without using Nakayama's lemma

I am trying to understand Nakayama's lemma. It looks like some "fixed point theorem". Using Nakayama's lemma , I can easily solve the following question. I want another proof. Thanks. Let $A$ be a ...
5
votes
2answers
277 views

Going up theorem (basic question)

If $S \subset R$ are commutative rings with $1$ and $R$ is an integral extension of $S$ then they have the same dimension. Basically the proof uses the going up theorem. But I have a question about ...
4
votes
1answer
455 views

Finite ring extension and number of maximal ideals

I want to understand why the following is true: Let $S \subseteq R$ be commutative rings with $1$ and assume that $R$ is finitely generated as an $S$-module by at most $k$ elements. For every ...
7
votes
2answers
663 views

Does the following property of the direct limit of a direct system follow from the axioms for a direct limit?

Question: Does it follow from the axioms for a direct limit that if $\mu_i(x_i)=0$ then there exists $j \geq i$ such that $\mu_{ij}(x_i)=0$? Definitions and notation: (Atiyah MacDonald, chapter 2, ...
3
votes
2answers
165 views

Noetherian ring and the same prime divisors

Let $A$ be a Noetherian ring and let $x \in A$ be an element which is neither a unit nor a zerodivisor. Why the ideals $xA$ and $x^{n}A$ (where $n \in \mathbb{N}$) have the same prime divisors? i.e. ...
3
votes
2answers
653 views

Hilbert's Nullstellensatz: intersection over maximal ideals?

In Reid's commutative algebra, the author gives some exposition about the Nullstellensatz, $I(V(J))=\operatorname{rad}(J)$. But I can't understand it: The Nullstellensatz says that we can take the ...
5
votes
4answers
4k views

Can you construct a field with 4 elements?

Can you construct a field with 4 elements? can you help me think of any examples?
15
votes
4answers
847 views

Why is ideal more important than subring?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
4
votes
2answers
576 views

Help understand canonical isomorphism in localization (tensor products)

Let $M,N$ be $A$-modules and let $P$ be a prime ideal. Can someone please explain why the following isomorphism holds? $$(M \otimes_{A} N)_{P} \cong M_{P} \otimes_{A_{P}} N_{P}$$ Here's what I ...
5
votes
4answers
1k views

Examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical examples. ...