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Results tagged with ra.rings-and-algebras
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user 9672
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.
2
votes
Maximal centralizer in full matrix ring
The answer is no.
Consider the matrix:
$$
\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
\end{pmatrix}
$$
It's centralizer algebra has dimension $8$, which is l …
- 5,654
13
votes
1
answer
697
views
Counting representations of $k[x,y]$ when $k$ is finite
$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n( …
- 5,654
8
votes
Centralizer of a Matrix over a Finite Field
Treat $F^n$ as an $F[t]$-module $M^A$, where $t$ acts by the matrix $A$. Then the centralizer can be thought of as $\mathrm{End}_{F[t]} M^A$. Now, $M^A$ has a primary decomposition
$ M^A = \bigoplus_ …
- 5,654
1
vote
On similar matrices and polynomial matrices
This proof is different from the one in Denis Serre's book.
As usual, take $M^A$ and $M^B$ to be the $k[t]$-modules with underlying space $k^n$, where $t$ acts by $A$ and $B$ respectively. Then $A$ i …
- 5,654
2
votes
Automorphisms of a matrix in Smith normal form?
If you have $P$, I think you can recover $Q$ as $(D^{-1}PD)^{-1}$.
Therefore, you are looking for invertible integer matrices $P$ such that $D^{-1}PD$ is also invertible (i.e., $P\in GL_n(\mathbf Z)\c …
- 5,654
15
votes
1
answer
1k
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 un …
- 5,654
40
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 ( …
- 5,654
2
votes
3
answers
266
views
An algebra constructed from symmetric differences
Let $S$ be a finite set. Let $R$ be a complex vector space with basis indexed by subsets of $S$. Define a product on $R$ by defining it on the basis elements as $1_A\cdot 1_B=1_{A\Delta B}$, where $A\ …
- 5,654