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Results tagged with ra.rings-and-algebras
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user 15934
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.
5
votes
Algebraic theorems with no known algebraic proofs
Okninski proved that $M$ is a monoid and $K$ is a field of characteristic $0$ such that the monoid algebra $KM$ is von Neumann regular, then $M$ is locally finite using analytic methods. No algebraic …
- 38.6k
3
votes
Examples of combinatorial problems where the only known solutions, or most "natural" solutio...
Many special cases of the Černý conjecture in automata theory have been proved using linear algebra, or really representations of monoids, that do not have known combinatorial proofs. The conjecture …
- 12.9k
7
votes
Accepted
True or false? Every left or right cancellative, duo semigroup is cancellative
Here is a right cancellative duo monoid that is not cancellative.
Let $A$ be a free abelian group of countable rank with basis $e_0,e_1,\ldots$. Let $\Phi\colon A\to A$ be the endomorphism given by $ …
- 38.6k
4
votes
If a semigroup embeds into a group, then is it a subdirect product of groups?
I seem to be late to this game, but since you asked for a reference request here is one which also gives a more elementary example which has interesting properties in its own right. The reference is P …
- 38.6k
14
votes
Accepted
Pairs of matrices for which traces of powers are independent of the order
If you look at the book of Radjavi on Simultaneous Triangularization it is shown that a semigroup of complex matrices which satisfies your trace property is simultaneously triangularizable, and so the …
- 38.6k
3
votes
What is known about finite dimensional modules over the nilCoxeter algebra?
@DaveBenson, has already given a beautiful answer to this question. I just wanted to point out that a number of the things he says (although not his full computation of the dimension of the Ext-algeb …
- 38.6k
3
votes
Accepted
Every homomorphism between (rational) Puiseux monoids is multiplication by a non-negative ra...
This seems to be Corollary 4 of the paper Morio Sasaki, Takayuki Tamura, Positive rational semigroups and commutative power joined cancellative semigroups without idempotent
Czechoslovak Mathematical …
- 38.6k
3
votes
Accepted
Minimal ideals and subalgebras of semisimple algebras
For a not necessarily unital ring $R$, a left $R$-module $S$ is simple if $RS\neq 0$ and $S$ has no proper submodule. A simple right module is defined dually. For a not necessarily unital ring the f …
- 38.6k
3
votes
Accepted
How is the classification of groups extensions by $H^2$ related to Yoneda Ext?
As I mentioned in my comment above, Gruenberg gave a direct bijection between $\operatorname{Ext}^1_{\mathbb ZG}(I_G,A)$ and group extensions $1\to A\to H\to G\to 1$. The details can be found in eith …
- 12.9k
5
votes
Congruences that aren't "finite from above"
On a ring or group all congruences are parafinite. Partition into the kernel and the complement of the kernel.
- 38.6k
10
votes
Accepted
An algebra with more than one Frobenius algebra structure
If $A$ is a Frobenius $K$-algebra and $\lambda\colon A\to K$ is a Frobenius form, then the Frobenius forms are the mappings of the form $a\mapsto \lambda(ua)$ with $u$ a unit of $A$.
One way to see th …
- 38.6k
1
vote
Accepted
Semigroup algebras with one dimensional center
Let me answer what goes on with bands. I'll assume the band is a monoid for simplicity. I think everything works so long as the band is unital but this can get messier to describe. Details of wha …
- 38.6k
10
votes
Accepted
Do rational group algebras have an outer automorphism?
This conjecture was proved in Feit, Walter; Seitz, Gary M.
On finite rational groups and related topics.
Illinois J. Math. 33 (1989), no. 1, 103–131. The proof relies on the classification since this …
- 38.6k
6
votes
Accepted
When is semigroup algebra local?
Question 1 either has a trivial answer or it is answered in my paper (with coauthors). REPRESENTATION THEORY OF FINITE SEMIGROUPS,
SEMIGROUP RADICALS AND FORMAL LANGUAGE THEORY depending on what you …
- 38.6k
1
vote
Hopf algebra of representative k-valued functions of an abstract group
This question is not well formulated because you don't define $R(G)$. Also it is not clear to me what you want to know,since your "if" clause has no "then" clause.
I assume you mean the Hopf algebra o …
- 38.6k