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

learn more… | top users | synonyms (1)

4
votes
2answers
661 views

Example of a non-Noetherian complete local ring

I was looking for an example of a non-Noetherian complete local commutative ring with $1$. I would appreciate if anyone can point to a reference.
3
votes
1answer
717 views

Definition of Ideals generated by a set

I'm struggling to understand the definition of ideals in ring homomorphisms generated by a set. If $R$ is commutative and has a $1$, then Ideal of $R$ generated by a subset $A$ of $R$: $$⟨ A ⟩ = ...
7
votes
3answers
174 views

Every element is radical in a field extension.

Let $L/K$ be an algebraic field extension. Suppose for each $x\in L$, there exists an integer $n>0$ such that $x^n\in K$, where $n$ may depend on $x$. If the characteristic of $K$ is zero, does it ...
3
votes
1answer
117 views

An algebraic result corresponding to etale morphism

This is an algebraic result corresponding to etale morphism which I want to prove: Let $k \to R$, $k$ is a field and $R$ is a local ring which is a finitely generated $k$-algebra, suppose the module ...
5
votes
1answer
210 views

How to view set of equivalence classes of extensions of M by N as an A-module

I know that for a commutative ring $A$ and $A$-modules $M$ and $N$, the set $E_A(M, N)$ of extensions of $M$ by $N$ can be equipped with the Baer sum which gives it an additive group structure. ...
0
votes
1answer
102 views

Extending an exponential valuation to the completion of a field

I was reading the section on Completions in Neukirch's Algebraic Number Theory. Neukirch uses the term multiplicative valuation for what other authors seem to call absolute value. He uses the term ...
4
votes
0answers
231 views

Noetherian rings, why commutativity?

I am looking for an answer to why one has to assume commutativity of a ring $R$ in proving some results about Noetherian rings. For example, Let $R$ be a commutative ring; look at the proof(s) of the ...
11
votes
2answers
266 views

$A\subseteq B\subseteq C$ ring extensions, $A\subseteq C$ finite/finitely-generated $\Rightarrow$ $A\subseteq B$ finite/finitely-generated?

Let $A\subseteq B\subseteq C$ be commutative unital rings. Recall that the extension $A \subseteq B$ is finite / of finite type / integral, when $B$ is a finitely generated $R$-module / when $B$ is a ...
2
votes
1answer
274 views

Rings such that $A[x]$ is a principal ideal domain

Let $A$ be a commutative ring. Then the following assertions are equivalent. $A$ is a field; $A[x]$ is a Euclidean domain; $A[x]$ is a principal ideal domain; $A[x]$ is a unique factorization ...
2
votes
1answer
215 views

Cancelling summands in a direct sum decomposition

Let $M$ be a Noetherian and Artinian module. Suppose that: $$\bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{t} B_{i} \cong \bigoplus_{i=1}^{q} A_{i} \oplus \bigoplus_{i=1}^{r} C_{i}$$ where all ...
3
votes
2answers
138 views

Rank of Noetherian modules

Suppose $M$ is a Noetherian left-module, why is the rank of $M$ unique? that is if $M^{r} \cong M^{s}$ as left modules then why $r=s$? Is this true if $M$ is Artinian?
3
votes
1answer
147 views

Noether Normalization in $\mathbb{C}[[x_1,…,x_n]]$

I have a problem in understanding the proof of the following theorem: Let $I\subseteq\mathbb{C}[[x_1,...,x_n]]$ be an ideal. Then there exists a $k\in\mathbb{N}$ and a linear coordinate change ...
4
votes
1answer
189 views

Submodules of Free modules counter-example: $Z/4Z$

Is it possible to use the ring $R=Z/4Z$ to construct a counter-example that submodules of free modules are not necessarily free? Thanks a lot.
1
vote
1answer
175 views

comaximality of ideals in a commutative ring with unit

Suppose we have a commutative ring $R$ with unit. I'm curious about what condition(s) on $R$ would be sufficient (without Axiom of Choice) to give a converse to the following familiar result: (#) If ...
2
votes
2answers
192 views

No ring isomorphism between certain rings

Let $k$ be an algebraically closed field and let$c,d$ be distinct elements of $k$. Why there is no ring isomorphism between $k[x,\frac{1}{x}]$ and $k[x,(x-c)^{-1},(x-d)^{-1}]$? I guess one approach ...
4
votes
2answers
240 views

Projective dimension of the residue field of a noetherian local ring. [duplicate]

Let $R$ be a commutative Noetherian local ring with maximal ideal $\mathfrak m$. Is it true that the projective dimension of $R/\mathfrak m$ is finite knowing that its injective dimension is ...
7
votes
2answers
801 views

The radical of a monomial ideal is also monomial

I have problems with this: I need to prove that in the polynomial ring the radical of an ideal generated by monomials is also generated by monomials. I found a proof on internet that uses the ...
1
vote
0answers
115 views

find all the polynomials that vanish in a variety.

Consider the equations $ x^3=0 , y^3=0 , xy(x+y)=0 $ where the polynomials live in $K[x,y]$ where K is a k-algebra (k field). Let V be the points that vanish on all this polynomials. Consider the ...
22
votes
2answers
2k views

Tensor products commute with direct limits

This is Exercise 2.20 in Atiyah-Macdonald. How can we prove that $\varinjlim (M_i \bigotimes N) \cong (\varinjlim M_i) \bigotimes N$ ? Atiyah gives a suggestion, he says that one should obtain a map ...
0
votes
1answer
394 views

radical of an ideal in the polynomial ring $k[x,y]$

How can I compute the radical of an ideal? I suppose that there no exist an algorithm for compute it. But in the case of polynomials rings? there exist an algorithm? I need to compute the radical of ...
3
votes
2answers
121 views

Finding a subring of the reals isomorphic to $\mathbb{Z}[t]/(4t+3)$

Let $I=(4t+3)$ be an ideal in $\mathbb{Z}[t]$. Find a subring of $\mathbb{R}$ isomorphic to $\mathbb{Z}[t]/I$. If $(4t+3)$ were monic, this question would be easily answered but since it isn't I'm ...
1
vote
1answer
123 views

Module isomorphic to a flat module

Let $M$ be a flat $A$-module, and $N$ a $A$-module isomorphic to $M$, what can we say about the flatness of $N$?
9
votes
3answers
285 views

Zariski topology in the complex plane: an example

I want to find the closure under the zariski topology, of this set $ \left\{ {\left( {x,y} \right) \in {\Bbb C}^2 ;\left| x \right| + \left| y \right| = 1} \right\} $ I have no idea what I can do
2
votes
1answer
80 views

polynomial ring, and some kind of algebraic number over the ring.

Let $k$ be a field, consider the ring $ k[X,Y]/(X^2-Y^3) $ I was proving something but I need to prove the existence of an element in the ring of fractions of $ k[X,Y]/(X^2-Y^3) $ such that satisfy a ...
11
votes
2answers
2k views

Show $\mathbb{Z}[\sqrt{6}]$ is a Euclidean domain

I'm attempting to modify the proof the $\mathbb{Z}[\sqrt{2}]$ is a Euclidean domain to prove a similar result for $\mathbb{Z}[\sqrt{6}]$. The idea is to prove that $\mathbb{Q}[\sqrt{6}]$ is Euclidean ...
7
votes
1answer
652 views

The Krull dimension of a module

Let $R$ be a ring, $M$ is a $R$-module. Then the Krull dimension of $M$ is defined by $\dim (R/\operatorname{Ann}M)$. I can understand the definition of an algebra in a intuitive way, since the ...
1
vote
0answers
105 views

Do modules have any topology?

Is there any kind of topology, natural or unnatural, that modules do have? Is there any geometric interpretation for flat modules? Is "exactness" of a sequence, any kind of geometric condition? ...
2
votes
2answers
93 views

How can one see that $\operatorname{tr}(f\otimes g)=\operatorname{tr}f\operatorname{ tr }g$?

Suppose you have two free modules $M$ and $N$ of finite rank over a commutative ring $R$. Let's also take some $f\in\operatorname{End}_R(M)$ and $g\in\operatorname{End}_R(N)$, which gives a ...
11
votes
2answers
270 views

Why is it that $\det(\phi-x\text{id})=\sum_{i=0}^n (-1)^ic_ix^i$?

I'm trying to understand a certain formula for the determinant in a more general setting. Say you have a free module $M$ of rank $n$ over a (commutative) ring $R$. Let ...
3
votes
2answers
284 views

How to find Krull dimension of $k[[x,y]][x^{-1},y^{-1}]$ where $k$ is a field?

I don't know how to find Krull dimension of $k[[x,y]][x^{-1},y^{-1}],$ where $k$ is a field?
2
votes
1answer
115 views

on generators of $k$-algebra

Let $(A,m,k)$ be a local ring, and $A$ is a finitely generated $k$-algebra, and the maximal ideal $m$ is nilpotent. Let $x_1,\ldots,x_n \in m$ and their canonical images in $m/m^2$ generate this ...
15
votes
1answer
279 views

A space of ideals

Definition: Let $R$ be a commutative ring with 1. Endow the power set $2^R$ with the product topology. The ideal space $\mathcal{I}(R)$ is defined to be subset of $2^R$ consisting of ideals, ...
12
votes
2answers
637 views

Motivation behind the definition of flat module

Can someone explain what is the motivation behind the definition of a flat module? I saw the definition but I don't really know why it is important to work with these structures.
3
votes
1answer
214 views

Poincaré series and short exact sequences

For an additive function $\lambda$ and an exact sequence of modules $0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$, we have $\lambda(M_2) = \lambda(M_1) + \lambda(M_3)$ by ...
0
votes
1answer
57 views

Find the kernel of $A[Y]\to A_a$.

Given an $A$-algebra homomorphism $A[Y]\to A_a$ by sending $Y$ to $1/a$, where $a$ is an element of $A$. We want to find the kernel $I$. The kernel $I$ is $(aY-1)$ . It is easy to see that the map ...
3
votes
1answer
197 views

Transcendental degree and dimension

I do not fully understand the proof of Lemma 5.6 in the book A Course in Commutative Algebra of Gregor Kemper (you can find it here) The lemma states that : If $A$ is an algebra over a field $k$, ...
4
votes
1answer
213 views

Corollary 2.13 of Atiyah - Macdonald

I just started learning about tensor products and I have some trouble understanding this corollary in Atiyah - Macdonald. All modules are assumed to be $A$ - modules for $A$ a commutative ring. ...
1
vote
1answer
243 views

A question about a proof of the Hilbert basis theorem

I don't understand a step in this proof of the Hilbert Basis Theorem. Here is the proof Planeth Math. I don't understand why $ \mathrm{deg} (f_{N+1}-g)< \mathrm{deg}(f_{N+1}) $. This can only ...
5
votes
1answer
189 views

Genus of the desingularization of a plane curve

Background I have been considering the following question. Let $k$ be an algebraically closed field and consider a curve $C \subset \mathbb{P}^2$. Compute its genus, that is, the genus of its ...
1
vote
1answer
193 views

Conceptualization of exterior powers of projective modules

Let $A$ be a commutative noetherian ring, and $P$ a projective $A$ module with $rank(P)=n$. I know that $\wedge^nP \simeq L$ for some rank 1 projective $A$-module, $L$; but I'm not sure of how to ...
2
votes
2answers
348 views

Is there an analogue of the jordan normal form of an nilpotent linear transform of a polynomial ring?

Is there an analogue of the Jordan Normal Form for an infinite dimensional vector space? In general I think the answer is no. It's been awhile since I studied it, but I'm pretty sure something would ...
3
votes
1answer
238 views

Minimal generating sets for homogeneous polynomial ideal in two variables

This question is (somehow) related to System of generator of a homogenous ideal Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let $$ {\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...
4
votes
1answer
138 views

Algebraic extension and Integral extension

If $K$ is algebraic closure of $F$, then as a ring, $K$ is integral over $F$. Is that true or not?
2
votes
1answer
149 views

A question with an odd hypothesis.

Let $S$ be a discrete valuation ring and $R\subset S$ be a proper subring (also a DVR). Assuming that $M$ and $N$ are the respective maximal ideals of $R$ and $S$ and that $N\cap R = M$, then the ...
9
votes
2answers
416 views

Exactness of sequences of modules is a local property, isn't it?

It's well known, that passing to modules of fractions is exact, i.e. if $M'\xrightarrow{f} M\xrightarrow{g} M''$ is an exact sequence of $A$-modules ($A$ being a commutative ring with ...
4
votes
1answer
138 views

Change of base property for flat modules?

I've read the claim about base change for flat modules in several sources (Lang's Algebra, Hartshorne's Algebraic Geometry, A&M), but unfortunately it isn't proven anywhere. The claim is that ...
2
votes
1answer
187 views

Does codimension equal height in complete local domains?

For an ideal $I$ in a commutative ring $R$, define $\operatorname{codim}I=\dim R-\dim R/I$. Does codimension equals height for all ideals in the formal power series ring? Does this hold for complete ...
5
votes
2answers
176 views

Prime ideals in certain local ring extensions

$(R,m)\subseteq (S,n)$ is a local extension of rings and $S$ is a finitely generated $R$-module. If $P$ is a prime ideal of $R$ such that $P\subset m^2$ and $P'$ is a prime ideal in $S$ such that ...
9
votes
2answers
439 views

Computing the “lying over”, “going up”, “going down” ideals.

For any commutative unital ring $R$ and an ideal $\mathfrak{a}$ of $R$, we shall denote $$\begin{align*} \mathrm{Spec}(R)&:=\{\text{prime ideals of }R\},\\ ...
7
votes
1answer
212 views

If $P$ is a prime ideal in a commutative Noetherian local ring $R$, is $P\hat{R}$ a prime ideal in $\hat{R}$?

Do prime ideals expand to prime ideals in the completion? I believe this is the case since I think $R/P\equiv \hat{R}/P\hat{R}$, although Atiyah-Macdonald explicitly mentions the preservation of ...