Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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44
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2answers
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 ...
35
votes
1answer
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 ...
39
votes
5answers
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 ...
0
votes
2answers
248 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 ...
13
votes
8answers
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 ...
13
votes
1answer
851 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 ...
13
votes
2answers
1k 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 ...
12
votes
1answer
720 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 ...
15
votes
3answers
814 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
1answer
631 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 $m$ ...
0
votes
1answer
233 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
2answers
302 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
1answer
339 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 $f$ ...
49
votes
16answers
7k 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 ...
64
votes
11answers
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 ...
25
votes
16answers
22k 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 ...
26
votes
6answers
5k 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 - ...
13
votes
4answers
1k 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 ...
18
votes
6answers
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 ...
10
votes
2answers
460 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, ...
21
votes
6answers
825 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 ...
11
votes
3answers
2k 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
1answer
527 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 ...
5
votes
1answer
1k 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. ...
17
votes
6answers
10k 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 ...
14
votes
3answers
540 views

Polynomial Rings

Let $R$ and $S$ be non-zero rings with identity. Is it possible to have $R[x] \cong S[[x]]$ ?
12
votes
1answer
615 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 ...
11
votes
1answer
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 ...
7
votes
2answers
735 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
4answers
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 ...
12
votes
5answers
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}: ...
11
votes
1answer
549 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, ...
6
votes
2answers
673 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 ...
5
votes
5answers
478 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
3answers
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
1answer
535 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. ...
2
votes
0answers
326 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 ...
13
votes
3answers
796 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
2answers
443 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 ...
10
votes
5answers
585 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 ...
9
votes
6answers
1k 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 ...
14
votes
1answer
499 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 ...
10
votes
2answers
381 views

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 ...
9
votes
2answers
258 views

Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$? To prevent things from being too easy, I ...
9
votes
0answers
460 views

Is “being a full ring of quotients” a Morita invariant property?

Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
8
votes
1answer
336 views

Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...
8
votes
1answer
1k views

Do these matrix rings have non-zero elements that are neither units nor zero divisors?

First, a disclaimer: This is a repost of a question I asked on stackexchange (no answer there). Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with ...
4
votes
1answer
354 views

Homotopy of quivers

The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows ...
4
votes
2answers
439 views

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 ...
3
votes
0answers
159 views

Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now. In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...