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Results tagged with ra.rings-and-algebras
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user 4213
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.
25
votes
Commutative subalgebras of M_n
See the related question at
Dimension of subalgebras of a matrix algebra .
In particular, I'd recommend the reference:
M. Mirzakhani `A simple proof of a theorem of Schur'
Amer. Math. Monthly 105 (199 …
- 20.8k
16
votes
Accepted
Semisimple-ish rings!
By Zorn's lemma, each right ideal is contained in a maximal right ideal,
therefore if $I+J = R$ then $I+M = R$ where $M$ is a maximal right ideal.
If $I+M\ne R$ for all maximal right ideals $M$ then $ …
- 20.8k
8
votes
An algebra constructed from symmetric differences
This is the group algebra $\mathbb{C}G$ where $G$ is an elementary
Abelian group of order $2^n$ where $n=|S|$ (the elements of $G$ are your $1_A$).
Every group ring over $\mathbb{C}$ is isomorphic to …
- 20.8k
6
votes
Accepted
Special subalgebras of central simple algebras
You want to know which algebras $A$ are such that $A\otimes B$ is central
simple for some algebra $B$.
(All algebras and tensor products being over $F$.) If $Z(A)$
and $Z(B)$ are the centres of $A$ a …
- 20.8k
6
votes
Schroeder-Bernstein for Rings
This is not even true for fields. Let $E_1$ and $E_2$ be isogenous but not
isomorphic elliptic curves over $K=\mathbb{Q}$ or $k=\mathbb{F}_p$
for some prime $p$. Then the isogeny $E_1\to E_2$ and its …
- 20.8k
5
votes
Commutative subalgebras of M_n
This is a reply to Tom's reply. Let's stick to commutative subalgebras
of $M_n(k)$ where $k$ is algebraically closed. Let $A$ be a unital
commutative subalgebra of $M_n(k)$. Then $N=k^n$ is a faithful …
- 20.8k
3
votes
Accepted
Intersection of ideals in C*-algebra or even rings in general
In the most general form, for arbitrary ideals over rings, this
is false. In the ring $\mathbb{Z}$ let $I_k$ be generated by $2^k$
and let $J$ be generated by $3$. Then $I_k+J=\mathbb{Z}$
for all $k$ …
- 20.8k
3
votes
$K_{0}(R) =\mathbb{Z}$ but some f.g. projective not stably free?
When $R$ is commutative, $K_0(R)$ is a commutative ring with
multiplicative unit the class $[R]$ of $R$. The only ring structure
with additive group $\mathbb{Z}$ is the familiar one, so every element
…
- 20.8k
3
votes
Understanding the modules of semiprimitive rings
You write
these simple modules can be
obtained by taking a quotient of the ring by maximal left ideals, and so these simple
modules are themselves also rings
but in general the quotient of …
- 20.8k
1
vote
Accepted
Calculating norms over a finite field (orthogonal groups).
It is the case that each isotropic vector in $V$ has the form
$u+w$ where $u\in U$ and $w\in W$ but $u$ and $w$ need not be isotropic.
To see where $2q-1$ and $q-1$ come from, the quadratic form on $ …
- 20.8k