Questions tagged [ra.rings-and-algebras]

Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

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39
votes
6answers
6k 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}: ...
80
votes
5answers
8k 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 ...
25
votes
2answers
1k views

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the differential operator corresponding ...
89
votes
2answers
6k views

$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$

Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
59
votes
3answers
6k 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 ...
23
votes
6answers
6k views

prime ideals in C([0,1])

It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa. So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
33
votes
1answer
2k views

Whence “homomorphism” and “homomorphic”?

Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” (and “homomorphism” as “property of ...
5
votes
1answer
213 views

Terminology for a monoid $H$ s.t. $xy \in H^\times$ only if $x, y \in H^\times$

The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of ...
92
votes
11answers
5k 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 ...
15
votes
1answer
900 views

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 ...
14
votes
1answer
2k views

Finite dimensional real division algebras

A celebrated theorem of Milnor and Kervaire asserts that any finite dimensional (not necessarily associative, unital) division algebra over the real numbers has dimension 1,2,4 or 8. This result is ...
6
votes
1answer
448 views

Is there an integral fusion ring which is not of Frobenius type?

Combinatorially, a fusion ring $\mathcal{F}$ is nothing but a finite set $B=\{b_1, \dots, b_r\}$ (generating the $\mathbb{Z}$-module $\mathbb{Z} B$) together with fusion rules: $$ b_i \cdot b_j = \...
2
votes
1answer
183 views

Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit

Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied? $$\text{(P) If }\,xy = x\...
105
votes
18answers
11k views

How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
31
votes
2answers
5k 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 ...
39
votes
5answers
4k views

When does a ring surjection imply a surjection of the group of units?

The following might be a very trivial question. If so, I don't mind it being closed, but would appreciate a reference where I could read about it. Let $R$ and $S$ be commutative rings and let $R^\...
23
votes
7answers
3k views

Commutative subalgebras of M_n

For a given $n$, is there any characterization for the commutative subalgebras of $M_n(\Bbb{C})$? I would like to know how many commutative subalgebras there are for each possible dimension. In view ...
13
votes
5answers
4k views

reduced ⊗ reduced = reduced; what about connected?

Several questions actually. All rings and algebras are supposed to be commutative and with $1$ here. (1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $...
31
votes
6answers
2k views

Bass' stable range of $\mathbf Z[X]$

Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication: ...
17
votes
3answers
2k views

Finite non-commutative ring with few invertible (unit) elements

for a ring $R$ with unity , let $U(R)$ denote the group of units of $R$ . Now there are lots of finite commutative rings, of arbitrarily high order, with exactly one unit ; indeed $U(R)=1$ for a ...
28
votes
1answer
2k 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
3answers
741 views

About the classification of commutative and of cocommutative, fin. dim. Hopf algebras

I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
16
votes
3answers
2k 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 ...
6
votes
1answer
954 views

Solid rings and Tor

A solid ring is a ring $R$ such that the multiplication $R\otimes_{\mathbb{Z}} R \to R$ is an isomorphism. These were classified by Bousfield and Kan; they are subrings of $\mathbb{Q}$, $\mathbb{Z}/...
11
votes
1answer
709 views

Does this algebra have finite global dimension ? (Human vs computer)

Usually computers can calculate the global dimension of a finite dimensional quiver algebra much faster than humans. But in this case a high end computer (calculating for 3 weeks) was not able to ...
4
votes
0answers
565 views

Rewrite sum of radicals equation as polynomial equation

My question is about a method described in Dr.Math forum for simplifying equations involving sums of radical functions. (The following is a transcription of the example given by Dr. Vogler): --- ...
2
votes
1answer
811 views

Are there infinitely many L-rigs?

$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to ...
0
votes
2answers
332 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 ...
107
votes
2answers
9k views

How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ 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 ...
62
votes
25answers
63k 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 ...
25
votes
8answers
5k 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 ...
45
votes
9answers
9k 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.
48
votes
7answers
12k 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 - ...
74
votes
1answer
4k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
19
votes
4answers
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 $\...
12
votes
1answer
4k 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. ...
15
votes
3answers
3k views

Which is the correct universal enveloping algebra in positive characteristic?

This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers. Let $\...
29
votes
1answer
2k 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 ...
17
votes
6answers
3k 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 ...
12
votes
1answer
1k 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, ...
22
votes
4answers
4k views

The number of ideals in a ring

Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here. Let $R$ be a finite commutative ring with identity. Under what conditions the number of ...
19
votes
4answers
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 ...
15
votes
3answers
551 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 require ...
15
votes
2answers
1k views

Which commutative groups are the group of units of some field?

Inspired by a recent question on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is ...
10
votes
1answer
623 views

How many Lie and associative algebras over a finite field are there?

This question is related to the following general question: Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
7
votes
2answers
3k views

*-homomorphisms between matrix algebras

Edited question: Are there any other non-trivial *-homomorphisms between matrix algebras apart from the unitary homomorphisms? Original question: Does there exist a surjective (but not bijective) *...
21
votes
3answers
1k 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 ...
17
votes
3answers
717 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
14
votes
1answer
882 views

Is the class of additive groups of rings axiomatizable?

I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which ...
12
votes
3answers
867 views

About enveloping algebras of direct sums

This question is imported from MSE. It is linked to this one in the case of semi-direct products. My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...

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