The commutative-rings tag has no usage guidance.

**8**

votes

**2**answers

599 views

### Is the support of an Artinian module finite?

Let $R$ be a commutative Noetherian ring, $M$ is an Artinian $R$-module. Is the set $Supp_R(M)$ finite?
Thanks.

**7**

votes

**2**answers

801 views

### Modules over Laurent series rings

Let $k[x]$ be the ring of polynomials over a field k in one variable x. A $k[x]$-module is a k-vector space together with a linear endomorphism (the action of x).
The field $k(x)$ of rational ...

**13**

votes

**2**answers

1k views

### Dimension 1 prime ideals in the intersection of two maximal ideals

This question/problem really comes from a fact in algebraic geometry, where it says that given an irreducible variety $V$ ($\dim V \geq 2$) then for any given pair of points $x,y\in V$ there is an ...

**1**

vote

**2**answers

709 views

### Rational power series

If we let $R=\mathbb{Z}[x]$ and $D=\mathbb{Z}[[x]]$. We say that $z\in D$ is rational if there is $g\in R$, $g\ne 0$ such that $zg\in R$. Let $S$ be the set of all rational elements in $D$. Then $S$ ...

**1**

vote

**2**answers

175 views

### Is there a relationship between the right global dimensions of R and R[1/v]?

A few days ago I asked a similar question about Krull dimension and got fantastic answers. Unfortunately, for the application I have in mind (a question on ring spectra), Krull dimension doesn't ...

**34**

votes

**3**answers

3k views

### What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...

**16**

votes

**4**answers

2k views

### Is there a Galois correspondence for ring extensions?

Given an ring extension of a (commutative with unit) ring, Is it possible to give a "good" notion of "degree of the extension"?. By "good", I am thinking in a degree which allow us, for instance, to ...

**7**

votes

**2**answers

867 views

### Is every poset the poset of prime ideals of a ring?

The answer to this question, as it is, is trivially false, for one necessary condition is the existence of maximal element(s), i.e., maximal ideals exist and are prime.
My question was inspired from ...

**9**

votes

**6**answers

2k views

### Applications of commutative algebra

Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think ...

**1**

vote

**1**answer

285 views

### A problem on Moebius transformations

We have the following result:
Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix ...

**2**

votes

**2**answers

639 views

### A problem for finite dimensional commutative algebra

Let $(A,m)$ be a local commutative associative algebra over the field of complex numbers, $m^n\ne 0$, $m^{n+1}=0$ for some $n>0$, and
(1) $A$ is finite dimensional as vector space
(2) for any ...

**11**

votes

**4**answers

929 views

### Are units of rings of functions on algebraic varieties finitely generated (mod. constants)?

Hello,
Consider the following question. Let $A$ be a finitely generated reduced algebra over an algebraically closed field $k$. Consider the group of units of $A$, modulo the group $k^*$. Is this ...

**20**

votes

**3**answers

1k views

### Invariance of $Z[x]$ under a self-equivalence of the category of commutative rings with 1.

Let $\mbox{Rings}$ be the category of commutative rings with $1$.
Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ s.t.
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$

**4**

votes

**1**answer

717 views

### module of differentials of formal power series ring and of its field of quotiens

For any $A$-algebra $B$ ( commutative ring with 1 ), we have the existence of $\Omega_{B/A}$, the module of relative differentials of $B$ over $A$, which can be defined by an universal property. In ...

**1**

vote

**1**answer

407 views

### Unimodular column property

Hi, I know that if $R$ is a ring such that every projective $R$-module finitely generated is free then $R$ has the unimodular column property.
I would like to know if there is a ring $R$ that doesn't ...

**7**

votes

**1**answer

728 views

### Direct sum of injective modules over non-Noetherian rings

Hi. I know, by the Bass-Papp theorem, that if every direct sum of injective $R$-modules is injective then $R$ is Noetherian. I would like to know if there exists a direct sum of injective $R$-modules ...

**1**

vote

**3**answers

566 views

### Stably free module not finitely generated is free

Hi. I have read that stably free modules not finitely generated are free; this is proved in
M.R. Gabel, stably free projectives over commutative rings, Thesis, Brandeis Univ., Waltham, MA 1972.
But ...

**12**

votes

**2**answers

2k views

### Generalized Euler phi function

Let $n$ be an integer, there is a well-known formula for $\varphi(n)$ where $\varphi$ is the Euler phi function. Essentially, $\varphi(n)$ gives the number of invertible elements in ...

**2**

votes

**2**answers

359 views

### Related to fractional ideals

$K$ a field, $A\subset K$ a subring, $M\subset K$ an $A$-submodule. Define
$$(A:_{K}M):= \lbrace s\in K|sM\subset A\rbrace$$
Then it is easy to see that
$$M\subset A\Longleftrightarrow A\subset ...

**6**

votes

**2**answers

974 views

### Is being finitely generated a local property?

I am trying ot figure out a proof of the following fact, that I believe is true, but it seems to me that something is lacking.
Suppose we have commutative, unitary rings $A,B$ and a (unit preserving) ...

**4**

votes

**0**answers

706 views

### Commutative ring Notes by M. Artin

In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...

**2**

votes

**0**answers

537 views

### Decomposition group vs Galois group of completed extension for height > 1 primes

Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...

**32**

votes

**7**answers

3k views

### Does $SL_3(R)$ embed in $SL_2(R)$?

Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$?
$SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...

**1**

vote

**0**answers

329 views

### Functoriality of a standard integral domain construction.

The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...

**5**

votes

**0**answers

984 views

### Are local, Noetherian rings with principal maximal ideal PIR?

A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider ...

**3**

votes

**0**answers

149 views

### Generic Rank of R^{1/p}

Suppose $R$ is a local Noetherian domain of dimension $d$ in characteristic $p>0$. Suppose $R^{1/p}$ is a finitely generated $R$-module, and suppose $k$ is the residue field of $R$. Is the ...

**1**

vote

**0**answers

495 views

### nilpotent matrices over polynomial rings

I am looking for an analogue of the Jordan normal form for nilpotent matrices over the
polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action ...

**4**

votes

**0**answers

219 views

### When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?

Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements ...

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votes

**0**answers

568 views

### Basic commutative algebra question.

Suppose that A is a local ring (commutative with unit), finite over a field k. Let L be the residue field A / m where m is the unique maximal ideal of A.
Does the dimension of L (as a k-vector space) ...

**2**

votes

**2**answers

350 views

### Homological dimensions of module

$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. ...

**1**

vote

**2**answers

764 views

### Existence of a minimal generating set of a module

Does a module (over a commutative ring) always possess a minimal generating set? When the module is not finitely generated, the typical Zorn's lemma type argument doesn't seem to work. More precisely, ...

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**4**answers

3k views

### Flatness and local freeness

The following statement is well-known:
$A$ a commutative Noetherian ring, $M$ a finitely generated $A$-module. Than $M$ is flat if and only if $M_{\mathfrak{p}}$ is free for all $\mathfrak{p}$.
My ...

**2**

votes

**2**answers

724 views

### Non-finite version of Nakayama's lemma?

Let $A$ be a local ring with nilpotent maximal ideal $\mathfrak{m}$ (i.e., some power of $\mathfrak{m}$ vanishes), and $M$ an $A$-module (not necessarily finitely generated). Let $\bar{S}\subset ...

**1**

vote

**4**answers

889 views

### Reduced rings, idempotents and their prime spectrum

Let $B$ be a commutative unitary reduced ring and let $A$ be a subring of it. Let $e$ be an idempotent of $B$. Then we have a natural surjective ring homomorphism $A\rightarrow Ae$ defined by ...

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votes

**1**answer

2k views

### On a theorem of Jacobson

In a comment to an answer to a MO question, in which Bill Dubuque mentioned Jacobson's theorem stating that a ring in which $X^n=X$ is an identity is commutative (theorem which has shown up on MO ...

**7**

votes

**1**answer

1k views

### Can an infinite commutative ring have a finite (but nonzero) number of non-nilpotent zero-divisors?

By a theorem of Ganesan, if a commutative ring not a domain has only finitely many zero-divisors, then the ring must be finite. (There are analogous results for non-commutative rings.)
There are ...

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votes

**3**answers

2k views

### What are the units in the ring of Laurent polynomials?

What are the units in $R[X,X^{-1}]$, where $R$ is a commutative ring with $1$? I know that the question for polynomial rings is a standard textbook exercise. However, I couldn't find a reference for ...

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votes

**1**answer

3k views

### A condition that implies commutativity

Let $R$ be a ring. A notable theorem of N. Jacobson states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for ...

**12**

votes

**4**answers

795 views

### What does the semiring of ideals of a ring R tell us about R?

Here is something I've wondered about since I was an undergraduate. Let $R$ be a ring (commutative, let's say, although the generalization to noncommutative rings is obvious). Ideals of $R$ can be ...

**9**

votes

**5**answers

2k views

### Axiomatic definition of integers

The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...