# Questions tagged [noncommutative-algebra]

For questions about rings which are not necessarily commutative and modules over such rings.

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### Every order in a finite-dimensional $\mathbb{Q}$-algebra is contained in a maximal order

Definition. A lattice in a finite-dimensional $\mathbb{Q}$-algebra $V$ is a finitely generated $\mathbb{Z}$-submodule $\mathcal{L} \subset V$ such that $\mathcal{L}\mathbb{Q}=V$ (i.e., $\mathcal{L}$ ...
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### Orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$. Throughout, we fix $F=\mathbb{Q}$. ...
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### Product of linearly dependent vectors is 0 in an anti-commutative algebra

From the 3rd edition of the book "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan, we find the following statement in chapter 4: Note that if ${v,w}$ is ...
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### How to determine if a non-commutative algebra is semisimple

Let $\cal{Q}$ be the field of rationals and $L=\cal{Q}(\sqrt 2, \sqrt 3)$ be a Galois extension of degree 4 ($[L:\cal{Q}]=4$). Using theory of Drinfeld twists on the galois group of $L$, I have ...
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### Why do we say that the tensor product of vector spaces is commutative, but the tensor product of vectors is not?

The Wikipedia article on the tensor product says The tensor product of two vector spaces $V$ and $W$ is commutative in the sense that there is a canonical isomorphism V ⊗ W ≅ W ⊗ V that maps v ⊗ w to ...
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### Definition of a basic algebra over a field $K$

Let $K$ be an algebraically closed field and let $A$ be a $K$-algebra with a complete set $\{e_1,…,e_n\}$ of primitive orthogonal idempotents. So, the algebra $A$ is called basic if e_iA \cong e_jA \...
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### Seeing that a product of quasi-free algebras is quasi-free in terms of lifting homomorphisms

A $k$-algebra over a field $k$ is quasi-free if for any bimodule $M$ the second Hochschild cohomology $H^2(A,M)$ vanishes or, equivalently, if for any square-zero extension $R/M=A$ there is a lifting ...
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### A generalization of the Clifford algebra

A minimal example of the Clifford algebra is the $\mathbb{C}$-algebra (unital, associative) generated by $x,y$ quotient over the relations \begin{eqnarray} x^2&=&1,\tag{1}\\ y^2&=&1,\...
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### Proof of (left) Ore condition implies there exists Q(R) a left quotient ring of R.

I am trying to understand the proof of Theorem 7.1.1 of `Noncommutative Rings' I. N. Herstein. Definition: An element in a ring is said to be regular if it is neither a left nor a right zero divisor ...
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### Request for reference books, articles, papers on the topic topic of Non Abelian Kummer Extensions.

I am a final year undergraduate student. I am doing my research on Non-Abelian Kummer Extensions. Can someone please introduce me to formal definition of Non-abelian Kummer Extensions. I know it is ...
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### Help me Understanding Non Abelian Kummer Extensions

[![I am doing my undergraduate research work on the Non-Abelian Kummer Extensions. I am following the book "Algebra" by Serge Lange. I have understood the difference between the abelian ...
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### Does Mashke's theorem hold for semisimple rings?

I saw a proof of Mashke's theorem using the theory of modules. However, it seems like it works in much more generality than it was stated in the text; can anyone confirm if it does, or if there's ... 1 vote
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### Number of solutions to simultaneous equations on lie groups

Say I have $n$ variables, special unitary operators from $SU(k)$, and write a set of $m$ equations that they must satisfy. These have the form $UVW...Z=I$, i.e. each one specifies that some product of ...
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### If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field. [closed]

I saw in Jacobson's book Finite Dimensional Division Algebras (1996) page 158 the sentence "If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F ...
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### Necessary and sufficient conditions for the function $f(t) := \mbox{trace}(A^{-1}e^{-tB} A e^{-tB})$ to be monotone decreasing.

Let $A$ and $B$ be a positive-definite $n \times n$ matrices. For any $t \ge 0$, define $f(t) := \mbox{trace}(A^{-1}e^{-tB} A e^{-tB}) = \|A^{-1/2}e^{-tB}A^{1/2}\|_F^2$. Question. What are necessary ...
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### division rings $D,D'$ both of finite dimension over their center $F$ are isomorphic as rings iff isomorphic as $F$-algebras?
Say I have two division rings, $D,D'$, both with center $F$ and both are of finite dimension over $F$, for some field $F$. Now suppose that $D\cong D'$ as rings, does if follow that $D\cong D'$ as $F$ ...