All Questions
Tagged with ra.rings-and-algebras reference-request
329 questions
16
votes
1
answer
1k
views
Lagrange's theorem for Hopf algebras
Under what conditions is a Hopf algebra free over any of its sub-Hopf algebras?
I am reading "Hopf algebras and their actions on rings" by Susan Montgomery, specifically chapter 3. Lagrange's theorem ...
- 85.6k
6
votes
2
answers
832
views
Constructing a ring from an abelian group in a minimal way
I'm looking for a specific construction, taking an abelian group (with designated element) $(G,+,1)$ to a commutative ring $(R,+,\cdot,1)$, where $G\subset R$ as a pointed abelian group, and which is ...
- 1,979
21
votes
4
answers
7k
views
Binomial Expansion for non-commutative setting
What could be a reference about binomial expansions for non-commutative elements?
Specifically, where can I find a closed formula for the expansion of $(A+B)^n$ where $[A,B]=C$ and $[C,A]=[C,B]=0$?
...
- 829
15
votes
1
answer
1k
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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 ...
- 5,654
5
votes
3
answers
1k
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adjoint of multiplication operator in a commutative algebra
Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
- 6,614
1
vote
1
answer
1k
views
Does it make sense that "Representations of groups over finite ring" ?
I am an undergrad student who wants to know about the representation theory over
arbitrary finite fields or finite rings of characteristic p (p a prime). (called modular
representation theory.)
In ...
- 11
10
votes
1
answer
1k
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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. ...
- 30.3k
3
votes
2
answers
590
views
Algebra with positive definite symmetrizing trace is semisimple.
This is a follow-up question to
When does a symmetric algebra over a field of characteristic 0 fail to be semisimple?
Let $H$ be a symmetric algebra over $\mathbb{R}$ with symmetrizing trace $\tau:...
- 1,021
9
votes
1
answer
1k
views
When does the homological dimension of a tensor product equal the sum of dimensions?
The notion of dimension I prefer most is right global dimension, but the question can also be asked for other notions (e.g. weak dimension, injective dimension, Krull dimension). Letting $d$ be ...
- 30.3k
4
votes
1
answer
382
views
cohomology of generalized Verma modules and invariant operators
First, let me fix some notation. Let $\mathfrak{g}$ be a semisimple Lie algebra and let $\mathfrak{p}$ be its parabolic subalgebra which induces the grading $\mathfrak{g} = {\mathfrak{g}}_{-} \oplus {\...
- 8,597
27
votes
1
answer
2k
views
Strong group ring isomorphisms
Background/Motivation
Let $R$ be a commutative ring with unit. If $G$ is a finite (or in general, discrete) group, let $R[G]$ be the group $R$-algebra associated to $G$. The isomorphism problem for ...
- 23k
4
votes
1
answer
497
views
Is the functor of divided powers a weakly monoidal functor?
Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\...
- 315
16
votes
2
answers
4k
views
A geometric reference for (affine) Gorenstein varieties and singularities
I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a ...
- 14.3k
5
votes
1
answer
618
views
Commutator formulas in a universal enveloping algebra
I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum ...
- 51
18
votes
4
answers
2k
views
Criteria for coherence of rings
I'm trying to collect pointers into the literature about coherent rings. Recall that a ring is left coherent if its finitely generated left ideals are finitely presented.
This condition was introduced ...
- 47.3k
6
votes
3
answers
3k
views
Defining Multiplication in Polynomials over Rings of Matrices
More explicitly, if $M_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), ...
- 370
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 ...
- 3,004
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
7
votes
1
answer
2k
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"Linear algebra" over Z/nZ - reference please!
Let A be a matrix with entries in Z/nZ. (n is not assumed to be prime.) Then the size of the row span is the size of the column span. All computations are mod n, so both these numbers are finite.
I ...
- 454
3
votes
0
answers
123
views
Name/references for analogue of ring with 2-cocycle condition instead of distributivity
I'm looking for a name for, and any past study on, the following kind of algebraic structure:
A set S equipped with an additive operation making it an abelian group, and a multiplication $*:S \times ...
- 7,320
24
votes
2
answers
3k
views
Does any textbook take this approach to the isomorphism theorems?
Below, I present an outline of a proof of the first isomorphism theorem for groups. This is how I usually think of the first isomorphism theorem for ______________, but groups will get the points ...
- 12.1k
4
votes
1
answer
305
views
Doing Real Algebraic Geometry on *-Rings
I've been searching google and scholar google, but i only have come upon orderings and Hermitian forms on *-fields.
Has real algebraic geometry been carried over to *-rings? *-rings are rings with an ...
user2529
31
votes
5
answers
5k
views
Gossip about Grothendieck and distributive lattices
In Gian-Carlo Rota's Indiscrete Thoughts, there a list of mathematical gossip among which one reads:
[...] What would have happened [...] if Grothendieck had known the theory of distributive ...
6
votes
2
answers
2k
views
Ideals in a noncommutative ring such that their product is their intersection?
If $R$ is a commutative ring and $I$ and $J$ are ideals in $R$ such that $I+J=R$ then $I \cap J=IJ$. This is not generally true in noncommutative rings, e.g. let $R$ be the lower triangular 2 x 2 ...
- 7,057
0
votes
0
answers
206
views
Vector-valued valuations on lattices
There's a fair amount of work on valuations on (modular) lattices, by which I mean functions $v : \mathcal{L} \rightarrow R$ that satisfy the modular expression
$$v(x) + v(y) = v(x \wedge y) + v(x \...
- 4,462
8
votes
0
answers
521
views
Skew polynomial algebra
When I was a very little hare, a big grey wolf told me about the following skew polynomial algebra, which I never understood. My question is whether the following construction is a part of some bigger ...
- 12.3k
15
votes
1
answer
633
views
Introduction to "commutative semialgebra"?
Of course, commutative algebra is a fundamental topic in algebraic geometry, number theory, representation theory, and so on.
However, there are some instances (most obviously tropical geometry) ...
- 12.6k
31
votes
11
answers
10k
views
Introduction to deformation theory (of algebras)?
So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
- 12.6k
19
votes
2
answers
1k
views
Hopf algebra reference
I was talking this morning to a colleague who thinks about combinatorial Hopf algebras. He mentioned several rings, which are of interest in combinatorics, for which he didn't know whether a Hopf ...
- 156k