The commutative-rings tag has no wiki summary.

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### A condition that implies commutativity

Let $R$ be a ring. A cute theorem by 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 the ...

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### 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 ...

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### Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
...

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### 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.
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### Are these rings of functions isomorphic?

Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are ...

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### 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 ...

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### Lengths over a local ring

Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$.
Is the following claim true?
Claim:
For any $\epsilon>0$, there ...

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votes

**6**answers

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### Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...

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### 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 ...

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**1**answer

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### Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...

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**2**answers

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### 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.

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### Maximal ideals of Z[x,y]

we know that the maximal ideals of ${\mathbb Z}[x]$ are of the form $(p, f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in ${\mathbb Z}[x]$ which is irreducible modulo $p$.
Is it true ...

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### $\lambda$-ring structure defined for a graded ring in Fulton-Lang's book

Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a special $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say ...

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### 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 ...

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### Quotient rings of $C(X)$

Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...

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### 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 ...

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### Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

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### Is the generation of rings by their units a question in K-theory?

Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:
Given an integral ...

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### Idempotent polynomials

Let $R$ be a commutative ring with identity and let $f \in R[x]$. There are well known characterizations for $f$ to be a nilpotent element of $R[x]$ or to have a multiplicative inverse in $R[x]$. Is ...

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### Periodic matrices

A square matrix $M$ such that $M^{k+1}=M$, for some positive integer $k$, is called a periodic matrix.
Can we characterize the periodic matrices in $\mathcal{M}_n(\mathbb{Z})$?
If we replace ...

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### 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 ...

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### 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 ...

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### 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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### Are roots of transcendental elements transcendental?

This looks extremely easy, but then again it's late at night...
Let $k$ be a commutative ring with unity. An element $a$ of a $k$-algebra $A$ is said to be transcendental over $k$ if and only if ...

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### Are there non-reflexive modules isomorphic to their bi-dual?

Let $M$ be an $R$-module. We say that $M$ is reflexive if the natural map $M\rightarrow M^{**}$ is an isomorphism.
I'd like to know if there exists a module isomorphic to its bi-dual but not ...

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### functional subrings

I should recall the notion of maximal subring of a commutative unitary ring $R$.
Def: A commutative ring $S$ is called a maximal subring of $R$ if $S \subset R$ and if $T \subset R$ constitute a ...

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### Isomorphic rings of functions

Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are ...

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### Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical?

I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero.
Definition. A ...

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**1**answer

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### Divisibility and factorization in rings that are not integral domains

In my course notes for an undergraduate course "Algebra I", I wrote at the point when I'm introducing the notion of divisibility in rings (in a section on unique factorization):
We want to study ...

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### Does existence of an isolated solution imply the Jacobian determinant is non-zero?

Let $f_1,\dots,f_n$ be formal power series in $\mathbb{C}[[x_1,\dots,x_n]]$ whose constant terms are all zero (i.e. $f_1,\dots f_n$ are not units in the ring). Suppose further that the radical of the ...

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### Maximal ideals of the rings of Baire- One Functions

A real function $f:X\rightarrow \mathbb{R}$ Is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...

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### Testing isomorphism of finitely generated algebras

Let $A=\mathbf{Q}[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over the rational numbers. Let $B=\mathbf{Q}[f_1,\ldots,f_r]$ and
$C=\mathbf{Q}[g_1,\ldots,g_s]$ be two finitely generated ...

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### 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 ...

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### When does End(M) consist entirely of zero, zero divisors, and units?

Let $R$ be a commutative ring (with $1$) such that every non-zero divisor in $R$ is a unit (see Rings in which every non-unit is a zero divisor for various stabs at what these are called). Let $M$ be ...

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### Subset of Spec(A) realized as inverse image of some Spec(B)

Let $A$ be a (commutative) ring and $U$ be an arbitrary subset of $\mathrm{Spec}(A)$. Do there exist a ring $B$ and a ring homomorphism $\varphi\colon A\to B$ such that ...

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### Homological characterization of smooth maps

Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings.
As far as I understand, in various generalizations of this situation, such a map is called smooth if $B$ ...

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### Is being finitely generated a local property?

Hi to everyone. 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 ...

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### Example of a reduced ring whose completion is not reduced

Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.

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### 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 ...

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### Algorithm to detect if an element of a (commutative) ring is in a subring?

For rings finitely-generated over a field, the theory of Groebner bases gives us quite an efficient algorithm for determining whether an element of the ring is in a given ideal of the ring.
Is there ...

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### Ext groups of affine scheme

Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...

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### 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, ...

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### 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 ...