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

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6
votes
2answers
303 views

(Minimal?) Polynomials using the Nullstellensatz

I'm struggling with an exercise that was asked in class: Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
1
vote
1answer
324 views

Standard graded algebra

I am so sorry if you feel this kind of question is not appropriate for MS. But I hope you can sympathize with me, I tried to find the answer in all my books and even Google but I found nothing. My ...
3
votes
2answers
217 views

About Gorenstein ring

Is it true that in a (non-local) Gorenstein ring, every maximal ideal has the same height? It seems a little strange, but I don't see any reason why it shoudn't.
6
votes
2answers
925 views

Why this element in this tensor product is not zero?

$R=k[[x,y]]/(xy)$, $k$ a field. This ring is local with maximal ideal $m=(x,y)R$. Then the book proves that $x\otimes y\in m\otimes m$ is not zero, but I don't understand what's going on, if the ...
3
votes
2answers
449 views

Every endomorphism of a finitely generated module satisfies a polynomial equation.

I encountered the following very interesting proposition in Atiyah's and McDonald's Introduction to Commutative Algebra: Let $A$ be commutative ring with identity, $M$ a finitely generated ...
8
votes
3answers
694 views

Alternative construction of Direct Limit

The construction of the direct limit that I learned from Atiyah Macdonald is the following: Suppose we have a directed system $(M_i,\mu_{ij})$ of $A$ - modules and $A$ - module homomorphims over a ...
7
votes
1answer
1k views

A free submodule of a free module having greater rank the submodule

Let $R$ be a commutative ring, and let $N\leq M$ be $R$-modules. Then, suppose $M$ and $N$ are free over $R$, if $R$ is an integral domain, then -considering the fraction modules over the quotient ...
2
votes
1answer
2k views

Commutative Ring: Nilpotent elements closed under addition? [duplicate]

Possible Duplicate: The set of all nilpotent element is an ideal of R Given a commutative ring $R$ and two nilpotent elements $r$, $s$ there exists an $n \in \mathbb{N}$ such that $$ ...
5
votes
1answer
337 views

Injective module and Noetherian ring

In the book Abstract Algebra of J.Antoine Grillet there is a theorem as follows: A ring $R$ is left Noetherian if and only if every direct sum of injective left R-modules is injective The ...
1
vote
1answer
49 views

Doubt about completeness

May I refer you to page 3 of: http://www.math.iitb.ac.in/atm/caag1/balwant.pdf Proof that $\hat{M}$ is complete, where it says "We choose $n(m)$ such that $n(m+1) \geq n(m)$ for every $m$". ...
2
votes
2answers
1k views

nilpotent ideals [duplicate]

Possible Duplicate: The set of all nilpotent element is an ideal of R An element $a$ of a ring $R$ is nilpotent if $a^n = 0$ for some positive integer $n$. Let $R$ be a commutative ring, ...
8
votes
2answers
580 views

Ideal as a projective module

I am sorry, this may not be a good question here. I am looking a good reference about when the ideal $I$ of a given commutative ring $R$ (local or may not be local) with identity is a projective ...
6
votes
1answer
271 views

Automorphisms of $k[x_1,x_2,\dots,x_n]$ that fix $k$

Given a field $k$, consider the polynomial ring $k[x_1,x_2,\dots,x_n]$. Is it possible to find all the automorphisms of this ring that fix the field $k$?
0
votes
2answers
101 views

Problem on multiplicative subset

Let $R$ be a ring, $S$ is a multiplicative subset of $R$. $a$ is an arbitrary element of $S$. Should there be 2 element $b,c \in S, b, c \neq 1$ such that $a=b.c$? If not please give a counter ...
2
votes
1answer
178 views

Lemma on finite generation of algebras over a field

I saw this lemma in some lecture notes, there was no proof given nor a reference, only a statement that it can be found in any text-book on commutative algebra. I checked several but couldn't find it. ...
4
votes
1answer
156 views

Algebraic maps from products of affine varieties

I have a question which might be fairly elementary, but I could not find an answer in the literature yet. Any pointers are very welcome :) Let $X$, $Y$ and $Z$ be affine algebraic varieties. I have a ...
3
votes
1answer
324 views

Two questions about integral “splitting ring” extensions

We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains ...
3
votes
1answer
196 views

On Krull dimension of $M/(0 :_{M} \mathfrak{m}^t)$ module

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module. There is an non-negative integer $t$ such that $M/(0 :_{M} \mathfrak{m}^t)$ is finitely generated. Then $$\dim ...
9
votes
2answers
245 views

Why is the (-1)-th coefficient of $f^n f'$ equal to 0, without dividing by $n+1$?

Let $R$ be a commutative ring, and $n$ be a nonnegative integer. Let $f\in R\left[t,t^{-1}\right]$ be a Laurent polynomial in one variable $t$ over $R$ (this means a formal $R$-linear combination of ...
4
votes
2answers
167 views

Infinite many curves passing through finite points?

Let $R$ be a Noetherian domain of dimension two. Let $\mathfrak{m}_1,\mathfrak{m}_2$ be two disctinct maximal ideals of height two. Are there always infinitely many prime ideals contained in ...
3
votes
0answers
187 views

Is an irreducible element still irreducible under localization?

Suppose $R$ is a domain. We say an element $x\in R$ is "irreducible" if $x=yz$ implies that $y$ or $z$ is a unit or both are units. I want to know if an irreducible element is still an irreducible ...
1
vote
1answer
142 views

How to find the generic initial ideal?

Here is an example from Ezra Miller's book: Combinatorial Commutative Algebra, pp. 26-27. Let $f,g\in k[x_1,x_2,x_3,x_4]$ be generic forms of degree $d$ and $e$. The generic initial ideal of ...
4
votes
2answers
860 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
1k 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
178 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
123 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
262 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
115 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
253 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 ...
13
votes
2answers
295 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
307 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 ...
3
votes
1answer
319 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
147 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?
4
votes
1answer
161 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 ...
1
vote
1answer
193 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
202 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 ...
5
votes
2answers
294 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 ...
8
votes
2answers
1k 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
121 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 ...
24
votes
2answers
3k 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
566 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
131 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
127 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
331 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 ...
8
votes
1answer
872 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
99 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 ...
12
votes
2answers
315 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 ...