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Results tagged with ra.rings-and-algebras
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user 290
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.
18
votes
Are automorphisms of matrix algebras necessarily determinant preservers?
Here is a positive result. Every finite-dimensional algebra $A$ over a field $K$ has an intrinsic determinant, and in fact an intrinsic characteristic polynomial, which is preserved by all automorphis …
- 118k
5
votes
Accepted
Real matrix rings and associative hypercomplex numbers
It's not clear what you mean by "real matrix ring," which could either mean a ring of the form $M_n(\mathbb{R})$ or a real subalgebra of $M_n(\mathbb{R})$; if the latter, this is the same as saying "f …
- 118k
1
vote
Constructing an adjunction between algebras and differential graded algebras
($R$ is commutative, of course.)
This functor does exist but it is probably not the functor you're looking for, in particular it is not the de Rham algebra. It is very strange. Here is a description o …
- 118k
4
votes
Accepted
Relationship between units of a ring and primitive characters of the ring under addition
The rings $\mathbb{Z}/n$ are the only examples.
I assume that "primitive character" just means that it is faithful, or equivalently that it does not factor through a proper quotient; this is the meani …
- 13k
26
votes
Integer matrices which are not a power
I actually even struggle to find examples of primitives matrices in these groups.
Here is a relatively easy sufficient condition. If $M \in SL_n(\mathbb{Z})$ is the $k^{th}$ power of some other matr …
- 118k
7
votes
On the tree-ishness of magmas and the stringiness of groups
People have done lots of interesting works along these lines. This is a discussion that would be best had at a blackboard to facilitate easy drawing, but here is one version of the story among many. F …
- 118k
8
votes
Accepted
Uniqueness of infinite direct sum decomposition
Yes. A semisimple module $M$ is canonically isomorphic to
$$M \cong \bigoplus_i \text{Hom}_R(S_i, M) \otimes_{\text{End}(S_i)} S_i$$
where $\text{Hom}_R(S_i, M)$ is what you might call the multiplicit …
- 118k
16
votes
How many Lie and associative algebras over a finite field are there?
Bjorn Poonen addresses this question for commutative (associative, unital) algebras in The moduli space of commutative algebras of finite rank; asymptotically we have
$$q^{\frac{2}{27} n^3 + O(n^{8/3} …
- 118k
8
votes
Category of modules over an Azumaya algebra and the Brauer group
$k$-linear cocomplete categories admit a "tensor product over $\text{Mod}(k)$" (thinking of them as module categories over $\text{Mod}(k)$) and the only thing you need to know about it to answer this …
- 118k
42
votes
Accepted
Why are ring actions much harder to find than group actions?
First, I am not really sure what you mean by "it is hard to come across a general theory of ring actions." This is precisely module theory! If $f : R \to S$ is any ring homomorphism whatsoever, then c …
- 4,713
15
votes
Dual of a bimodule
As explained in more detail in this blog post linked by Jakob in the comments, every $(A, B)$-bimodule $M$ has two natural duals:
If $M$ is finitely generated projective as a left $A$-module, it has …
- 118k
8
votes
Is the multiplication between even numbers an associative algebra?
The set of even numbers is a non-unital ring, in particular it has an associative multiplication, but you are right that it isn't an algebra over a field. For an example of a non-unital algebra, cons …
- 4,713
30
votes
intuition for hochschild homology
Slogan: Hochschild homology is a (derived) categorification of the trace.
This means the identity at the end of John Pardon's answer is a categorification of the identity $\text{tr}(AB) = \text{ …
- 118k
17
votes
Accepted
Is there any "fundamental" distinction between min-plus, max-plus, min-product, and max-prod...
Consider the following four semirings, listed in the order underlying set, addition, additive identity, multiplication, multiplicative identity:
$A = (\mathbb{R} \cup \{ \infty \}, \text{min}, \inft …
- 118k
4
votes
Graded rings with compatible S_n actions
As a warmup, an $\mathbb{N}$-graded ring is a monoid object in the symmetric monoidal category of $\mathbb{N}$-graded abelian groups under the convolution tensor product, which you can think of as Day …
- 118k