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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42 votes
6 answers
7k 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}: ...
36min's user avatar
  • 3,748
88 votes
5 answers
10k 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 ...
Martin Brandenburg's user avatar
26 votes
2 answers
2k 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 ...
Ali Taghavi's user avatar
91 votes
2 answers
7k 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 $A \not\cong A \oplus \mathbb{Z}$?
Martin Brandenburg's user avatar
121 votes
18 answers
14k 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 ...
61 votes
3 answers
7k 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 ...
Johan Öinert's user avatar
33 votes
1 answer
3k 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 ...
Francois Ziegler's user avatar
25 votes
6 answers
7k 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, ...
Nikita Kalinin's user avatar
5 votes
1 answer
236 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 ...
Salvo Tringali's user avatar
3 votes
1 answer
1k 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 ...
Sylvain JULIEN's user avatar
95 votes
11 answers
6k 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 ...
Philipp Lampe's user avatar
15 votes
1 answer
1k 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 ...
Berry's user avatar
  • 269
14 votes
1 answer
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 ...
Adam Epstein's user avatar
  • 2,440
7 votes
1 answer
550 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 = \...
Sebastien Palcoux's user avatar
2 votes
1 answer
208 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\...
Salvo Tringali's user avatar
39 votes
5 answers
5k 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^\...
José Figueroa-O'Farrill's user avatar
33 votes
2 answers
7k 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 ...
François Brunault's user avatar
31 votes
6 answers
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: ...
Oblomov's user avatar
  • 2,501
28 votes
1 answer
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 "...
Steven Sam's user avatar
25 votes
7 answers
4k 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 ...
Carmen's user avatar
  • 387
18 votes
3 answers
3k 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 ...
user avatar
17 votes
3 answers
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 ...
David Roberts's user avatar
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12 votes
1 answer
5k 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. ...
Jose Brox's user avatar
  • 2,962
12 votes
5 answers
5k 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 $...
darij grinberg's user avatar
12 votes
3 answers
656 views

IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
Martin Brandenburg's user avatar
12 votes
1 answer
876 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 ...
Mare's user avatar
  • 25.7k
10 votes
0 answers
427 views

In search of lost graded rings

$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
Emily's user avatar
  • 10k
10 votes
3 answers
1k 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....
Konstantinos Kanakoglou's user avatar
6 votes
1 answer
1k 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}/...
Jeff Strom's user avatar
  • 12.5k
4 votes
0 answers
731 views

Rewrite sum of radicals equation as polynomial equation

My question is about a method described in [Dr.Math forum][1] for simplifying equations involving sums of radical functions. (The following is a transcription of the example given by Dr. Vogler): --- ...
mvc's user avatar
  • 153
0 votes
2 answers
355 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 ...
Suman's user avatar
  • 1,211
113 votes
2 answers
12k 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 ...
Bill Dubuque's user avatar
  • 4,706
74 votes
1 answer
5k 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 ...
Martin Brandenburg's user avatar
63 votes
25 answers
68k 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 ...
50 votes
7 answers
14k 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 - ...
47 votes
9 answers
11k 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.
39 votes
9 answers
10k views

Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of rings that are not isomorphic to their opposite rings? Is there a science to constructing them? The only simple example known to me: In Jacobson's Basic Algebra (...
Amritanshu Prasad's user avatar
36 votes
3 answers
2k views

Are large powers of polynomials linearly independent?

Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional. Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
Guillaume Aubrun's user avatar
34 votes
1 answer
5k views

Freyd-Mitchell's embedding theorem

Freyd–Mitchell's embedding theorem states that: if $A$ is a small abelian category, then there exists a ring R and a full, faithful and exact functor $F\colon A \to R\text{-}\mathrm{Mod}$. I have been ...
Bruno Stonek's user avatar
  • 2,914
34 votes
8 answers
4k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
26 votes
8 answers
6k 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 ...
Qiaochu Yuan's user avatar
22 votes
4 answers
5k 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 ...
alex alexeq's user avatar
  • 1,841
20 votes
3 answers
2k 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 ...
Anton Geraschenko's user avatar
20 votes
4 answers
3k 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 $\...
Peter Goetz's user avatar
19 votes
4 answers
2k 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 ...
Thomas Kahle's user avatar
  • 1,961
17 votes
6 answers
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 ...
jlk's user avatar
  • 3,234
17 votes
3 answers
855 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 ...
Nick S's user avatar
  • 1,990
16 votes
2 answers
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 ...
Qiaochu Yuan's user avatar
16 votes
0 answers
842 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 ...
Torsten Schoeneberg's user avatar
16 votes
3 answers
673 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 ...
zcn's user avatar
  • 686

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