**17**

votes

**2**answers

2k views

### Dimension of infinite product of vector spaces

This question is motivated by the question link text, which compares the infinite direct sum and the infinite direct product of a ring.
It is well-known that an infinite dimensional vector space is ...

**46**

votes

**2**answers

4k views

### How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric series. In ...

**36**

votes

**1**answer

2k views

### What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good ...

**41**

votes

**5**answers

3k views

### when is A isomorphic to A^3?

this is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...

**12**

votes

**1**answer

801 views

### finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional division algebra over the real numbers has dimension 1,2,4 or 8. This result is established using methods from algebraic ...

**0**

votes

**2**answers

260 views

### Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...

**14**

votes

**8**answers

3k views

### What is the “right” definition of a ring?

This is somewhat related to Greg's question about groups and abelian groups. Suppose you met someone who was well-acquainted with groups, but who was unwilling to accept rings as a meaningful object ...

**20**

votes

**9**answers

4k views

### What are the reasons for considering rings without identity?

I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.

**13**

votes

**1**answer

1k views

### Koszul duality between Weyl and Clifford algebras?

Koszul duality
Given a finite-dimensional $k$-vector space $V$ (I am happy taking $k = \mathbb{C}$ anywhere in the following if it makes a difference) and a subspace $R \subseteq V \otimes V$, we can ...

**18**

votes

**6**answers

12k views

### Determinant of sum of positive definite matrices

Say $A$ and $B$ are symmetric, positive definite matrices.
I've proved that $\det(A+B) \ge \det(A) + \det(B)$ in the case that $A$ and $B$ are two dimensional.
Is this true in general for ...

**12**

votes

**1**answer

613 views

### Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL_n(K)$. Therefore, ...

**15**

votes

**3**answers

914 views

### Characterising categories of vector spaces

Consider the category $FdVect_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only ...

**12**

votes

**1**answer

675 views

### Number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?

Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?
The number of idempotent matrices over a finite field is well-known and since we can decompose $m$ ...

**3**

votes

**2**answers

662 views

### Skew fields inside quaternion division algebras

Suppose that $Q$ is a quaternion division algebra with center $k$, where $k$ is an arbitrary commutative field (let's say with $\operatorname{char}(k) \neq 2$ if necessary). Assume that $D$ is an ...

**0**

votes

**1**answer

241 views

### Example of a ring satisfying this variant definition of “symmetric” on nilpotent elements

I want an example to show that if $a,b$ are nilpotent elements of a ring $R$ with 1 and if $c$ is any element of $R$, then $abc=0\Rightarrow acb=0$ but $cab=0$ does not imply $acb=0$.
This is unlike ...

**1**

vote

**2**answers

310 views

### A basic question about rings

Perhaps this is a trivial question, but I have no idea how to justify it.
Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and ...

**1**

vote

**1**answer

351 views

### The space $\psi$

Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ ...

**50**

votes

**16**answers

8k views

### Why is it a good idea to study a ring by studying its modules?

This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...

**62**

votes

**9**answers

4k views

### Can we unify addition and multiplication into one binary operation? To what extent can we find universal binary operations?

The question is the extent to which we can unify addition
and multiplication, realizing them as terms in a single
underlying binary operation. I have a number of questions.
Is there a binary ...

**31**

votes

**16**answers

27k views

### Linear Algebra Texts?

Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...

**64**

votes

**11**answers

3k views

### Can a non-surjective polynomial map from an infinite field to itself miss only finitely many points?

Is there an infinite field $k$ together with a polynomial $f \in k[x]$ such that the associated map $f \colon k \to k$ is not surjective but misses only finitely many elements in $k$ (i.e. only ...

**28**

votes

**6**answers

6k views

### Good lattice theory books?

A recent answer motivated me to post about this. I've always had a vague, unpleasant feeling that somehow lattice theory has been completely robbed of the important place it deserves in mathematics - ...

**20**

votes

**7**answers

2k views

### Consequences of not requiring ring homomorphisms to be unital?

As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...

**13**

votes

**4**answers

2k views

### Relationship between the cohomology of a group and the cohomology of its associated Lie algebra.

Let $G$ be a group and let $k$ be a field (characteristic 0 if you want). Let $L$ be the graded Lie ring associated to the lower central series of $G$, that is, $L$, as a graded abelian group is ...

**21**

votes

**6**answers

869 views

### How to recognize a Hopf algebra?

Suppose we are handed an algebra $A$ over a field $k$. What should we look at if we want to determine whether $A$ can or cannot be equipped with structure maps to make it a Hopf algebra?
I guess in ...

**13**

votes

**3**answers

3k views

### Why is a ring called a “ring”?

Why is a ring called "ring" (or Zahlring in German)? There seems to (naive) me nothing more ring-like to a ring than there is to a group or a field. I am particularly interested to learn why the ...

**10**

votes

**2**answers

499 views

### Exact sequence of monoids

What is the right definition of an exact sequence of monoid homomorphisms?
I can't seem to find a consistent in my searches; indeed Balmer (Remark 2.6,
...

**5**

votes

**1**answer

2k views

### What are tame and wild hereditary algebras?

What are tame and wild hereditary algebras?
Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. ...

**16**

votes

**3**answers

1k views

### does the “convolution theorem” apply to weaker algebraic structures?

The Convolution Theorem is often exploited to compute the convolution of two sequences efficiently: take the (discrete) Fourier transform of each sequence, multiply them, and then perform the inverse ...

**12**

votes

**1**answer

2k views

### Characterizations of UFD and Euclidean domain by ideal-theoretic conditions

This questions is inspired by an exercise in Hungerford that I have only partially solved. The exercise reads: "A domain is a UFD if and only if every nonzero prime ideal contains a nonzero principal ...

**16**

votes

**4**answers

1k views

### What is the geometric object corresponding to a subalgebra in a polynomial ring

Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...

**14**

votes

**3**answers

552 views

### Polynomial Rings

Let $R$ and $S$ be non-zero rings with identity. Is it possible to have $R[x] \cong S[[x]]$ ?

**13**

votes

**1**answer

657 views

### Are wild problems related to undecidable ones?

In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an ...

**10**

votes

**1**answer

550 views

### Why not _co_free modules?

Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...

**7**

votes

**2**answers

885 views

### Generalization of finitely generated, finitely presented modules?

Let $R$ be a commutative ring and $M$ an $R$-module.
The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$.
Similarly, $M$ is finitely presented iff there is an ...

**7**

votes

**2**answers

748 views

### What is the smallest $C^*$-algebra containing the “standard” pseudodifferential operators?

Is $\Psi^0(\mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|\partial^\alpha_x \partial^\beta_\xi a(x,\xi)| \leq C_{\alpha,\beta} (1+|\xi|)^{-|\beta|}
$
) a $C^*$-algebra?
In other ...

**20**

votes

**4**answers

1k views

### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

**15**

votes

**1**answer

884 views

### Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...

**12**

votes

**5**answers

1k views

### An algebra of “integrals”

When discussing divergent integrals with people, I got curious about the following:
Is there an $\mathbb{R}$-algebra $A$ together with a map (could be defined on just a subspace)
$$\int_0^{\infty}: ...

**12**

votes

**6**answers

2k views

### What's an example of a transcendental power series?

Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$?
I am looking for elementary example (so there should be a proof of transcendence that does ...

**5**

votes

**5**answers

659 views

### Noncommutative computational package

I am wondering if there is a program which can do simple operations over noncommutative rings, like expand products and substitute one expression for another.
To clarify, consider the following ...

**4**

votes

**3**answers

2k views

### Intuitive Example of a Jacobson Radical

Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand Wikipedia's explanation.

**3**

votes

**0**answers

343 views

### Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task:
I have two graded (generally ...

**3**

votes

**1**answer

569 views

### Is there a way to define a prime ideal object via diagrams in the category of rings?

I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...

**16**

votes

**1**answer

598 views

### Does GL_n(Z) have a noetherian group ring?

Has the (left, right, 2-sided) noetherian property of the integral group ring of arithmetic groups like $GL_n(Z)$ been considered in the literature?
Motivation: a recent trend has been to study ...

**14**

votes

**3**answers

821 views

### Can a module be an extension in two really different ways?

(Edit: I've realized that there was an error in my reasoning when I was convincing myself that these two formulations are equivalent. Hailong has given a beautiful affirmative answer to my first ...

**11**

votes

**5**answers

637 views

### Is complete homogeneous symmetric polynomials, an irreducibile element in Polynomial ring?

Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$.
$$ h_a=\text{ sum of all monomials of degree ...

**11**

votes

**2**answers

455 views

### Noncontractible connected topological rings ?

Are there any non-contractible connected topological rings?
Of course, such a thing cannot be a (topological) algebra over the reals.
(I have a vague memory of having a glance at an erticle by ...

**6**

votes

**2**answers

587 views

### Generalizing the Fundamental Theorem of Symmetric Polynomials

The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...

**16**

votes

**1**answer

935 views

### Two rings…are they isomorphic?

Edit: I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments.
Consider the local rings
$$R = \mathbb{C}[[x,y,z]]/\langle ...