# Questions tagged [division-algebras]

A division algebra $D$ is a vector spaces over a field $F$ equipped with a bilinear product and a multiplicative neutral element $1$. All the non-zero elements of $D$ have a multiplicative inverse. Associativity is often assumed but not always. Any field is a commutative, associative division algebra. A skewfield = a division ring is always a division algebra over its center. The quaternions form the best known non-commutative division algebra.

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### What happens if in algebra we have several elements whose reciprocal is zero? [closed]

Suppose, we have in an algebraic system $\omega_1\ne\omega_2\ne\omega_3...$ and $1/\omega_1=0$, $1/\omega_2=0$, $1/\omega_3=0$,... What consequences should we expect? Is there a term for an element ...
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### Divison algebra vs general/abstract algebra

I am not a mathematician but confused about division algebra vs. algebra. I suspect that "division algebra" is a sub-category(literally, not a math concept) of general or abstract algebra ...
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### Since each $M_n(D)$-submodule of $M_n(D)$ is also a $D$-submodule, we have $M_n(D)$ artinian as a ring.

$D$ is a division ring so it is artinian, hence so is $M_n(D)$ as a $D$-module. Since each $M_n(D)$-submodule of $M_n(D)$ is also a $D$-submodule, we have $M_n(D)$ artinian as a ring. I am not sure ...
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### Uncommon notation for division algebra

I have found the following notion for a division algebra in a paper. $K=\mathbb{R}(x_1, \dots, x_n)$ is the field of rational functions in $n$ variables over $\mathbb{R}$ and $F=K((t))$ be the field ...
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### Is every non-zero two-sided ideal of a polynomial ring over a division ring intersecting with the center?

Let $D$ be a division ring with center $F$, and $J$ a non-zero two-sided ideal of $D[x]$. Is it true that $J \cap F[x] \neq 0$? This is a question spawned from another problem I'm working on. And I ...
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### For every element of an associative division $\mathbb{R}$-algebra there exists a quadratic equation with that element as a solution

If $D$ is an $\mathbb{R}$-algebra that is also a division ring and $\dim_{\mathbb{R}}D=n<\infty$, then for every $d\in D$ there exists $\lambda\in\mathbb{R}$ such that $d^2+\lambda d\in\mathbb R$. ...
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### Bimodule over division algebras

Let $E_1$, $E_2$ be finite-dimensional division algebras over $\mathbb{Q}$. Let $X$ be a left $E_1 \otimes_\mathbb{Q} E_2^{op}-$module. In other words, $X$ is an $E_1-E_2-$bimodule, and $\mathbb{Q}$ ...
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### Finite dimensional skew fields over $\mathbb{Q}$

Is there a specific reference for finite dimensional, associative, unital $\mathbb{Q}-$algebras that are division rings? Or also more in general, the type of questions I am trying to look into are: ...
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### Normed Division Algebras: Why those dimensions? [duplicate]

I recently read a fascinating article about string theory, which discussed higher-dimensional algebras and their applications to supersymmery. The author mentioned that there were only four algebras ...
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### Decomposition of a rank-deficient matrix

Given a complex or real $n\times m$ matrix $M$ with rank $r$, one can write it as $$M=LR$$ where $L$ is a $n\times r$ matrix, and $R$ is a $r \times m$ matrix. Does this hold for arbitrary fields? (I'...
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### Does there exist a division ring without unity?

In abstract algebra I have only ever seen division introduced via multiplicative inverses, namely starting from a ring with unity $R$ and then adding the condition that each element $x$ has an inverse ...
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### If $\Bbb R^n$ has a commutative division algebra structure over $\Bbb R$ then the multiplication map on $\Bbb R^n$ is continuous

I was reading the proof of the following theorem in Hatcher's Algebraic Topology: Theorem 2B.5. $\Bbb R$ and $\Bbb C$ are the only finite-dimensional divison algebra structure over $\Bbb R$ and have ...
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### Are all central simple algebras unital?

The definition I'm using for a CSA over a field $k$ is the following: A CSA over $k$ is a finite-dimensional associative $k$-algebra which is simple and has center precisely $k$. My question ...
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### Maximal central subalgebra(s) in non-central division $k$-algebra

Let $D$ be a finite dimensional (non-central) division $k$-algebra, where $k$ is a field. Is there a concrete description of the maximal subalgebra(s) of D having center $k$?
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### Central simple algebra of dimension 4

Suppose $A$ is a $F$-central simple algebra with maximal subfield $E$ such that $[A:F] = 4$. if $N_{E/F}(E^*) \ne F^*$, then $A$ is a division algebra. Is this even true? If it is true how i can ...
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### Splitting fields of a divison algebra

Let $k$ be a field, $D$ be a central division algebra of degree $n$ over $k$. We call $k'$ a splitting field of $D$ if $D\otimes_kk'\cong M_n(k')$. Splitting fields may not be isomorphic, can we say ...
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### How do I prove that every fractional ideal of an order in a division algebra is a full lattice?

Let O be an R-order for some Dedekind domain R, let F be the field of fractions of R and D be a division algebra over F. A fractional left ideal of O is an R-lattice I in D such that OI in I (I ...
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### Are all quaternion algebras over the rationals skew fields?

If I understand correctly, any quaternion algebra over the rationals is a noncommutative associative division algebra. I am currently working with implementations of quaternion algebras in MAGMA and ...
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### Why division ring has a center of a ring (=subring) is commutative and therefore division ring reflect itself a field?

I thought a ring was commutative for another reason but I realized that something I had not yet discovered, had led me to look for the solution in the wrong place. I see that 'commutative' property of ...
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### Central simple algebras give rise to algebraic groups

Let $D$ be a central division algebra over a field $k$ of dimension $n^2$. I have heard that the functor $$R \mapsto (D \otimes_k R)^{\ast}$$ going from commutative $k$-algebras to groups is ...
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### Central simple quaternion algebra: why is the matrix for $\rho(v)$ antidiagonal?

Let $F$ be a field of characteristic $0$. Let $D$ be a central, simple quaternion division algebra over $F$. Let $x \in D$, not in $F$. Then $K = F[x]$ is a field of degree two over $F$, and $D$ is ...
Let $D$ be a division ring with the center $F$. Suppose that $G$ is a subgroup of the multiplicative group of $D$ such that every element of $G$ is algebraic over $F$. Then may we conclude that any ...
Let $F$ be a field of characteristic 2. How could we construct a division ring $D$ which centre is $F$. Where division ring mean non-commutative ring with unity $1$ and for each non-zero element \$x \...