# Questions tagged [algebras]

For questions about algebras, their properties, and their structures. Use [tag:algebra-precalculus] or [tag:abstract-algebra] if your question is about algebra, not algebras.

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### Bilinear form on a finite dimensional algebra over a field

I am reading the first chapter of 'methods of representation theory, Volume I' section 9A, written by Charles W.Curtis & Irving Reiner. Let $A$ denotes a finite dimensional algebra over an ...
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### spectrum of a general algebra

For unital Algebras $A$, the Spectrum of an element $x \in A$ is defined to be $$Sp_A(x) := \{ \lambda \in \mathbb{C}: x - \lambda * e \text{ is not invertible} \}$$ According to my textbook, there ...
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### Zero divisors in an algebra with two generators [closed]

Let $k$ be a field, and $R = k\langle x,y \mid x^2 = 0\rangle$. The elements $x$ and $y$ are not supposed to commute with each other. Is the only case where nonzero elements $a, b \in R$ satisfy $ab=0$...
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### Turning a subalgebra into an ideal by changing the algebra it is contained in

Let $B$ be an (not necessarily unital) algebra and $A \subseteq B$ a subalgebra. Is there some sort of quotient $C$ of $B$ (or any other interesting construction not necessarily found by taking ...
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### Semisimplicity implies separability for a perfect field

Let $k$ be a field, $A$ a finite-dimensional semisimple $k$-algebra. If $k$ is a perfect field (every finite field extension of $k$ is seperable), then $A$ is separable. I know a proof that uses the ...
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### Proof the quaternions are 4-dimensional?

The quaternions can be defined as $$\mathbb{R}\langle X,Y\rangle/(X^2+1,Y^2+1,XY+YX)$$ From these relations, it is relatively easy to prove that $1,X,Y,XY$ span the quaternions over $\mathbb{R}$. But ...
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### Inferring classification of Clifford algebras from classification of Clifford modules

Let $Cl_n$ be the Clifford algebra (over reals) $$Cl_n = T^{*}\mathbb{R}^n/\langle v\otimes v - q(v) \rangle.$$ There is a periodic table of $K$-representations of $Cl_n$, i.e. $\mathbb{R}$-linear ...
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### Equivalent definitions of Hopf algebras

Recently, I started to study the book Hopf algebras by Moss Sweedler, in such book, given a coalgebra $(C,\Delta,\epsilon)$ and an algebra $(A,\mu,\eta)$, the autor defines the convolution of two ...
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### Jordan-Holder theorem for group algebras

I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite ...
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### Definition of splitting field: Why do we require centrality?

Let $k$ be a field. Let $D$ be a division algebra over $k$. Call a field extension $K/k$ a splitting field for $D$ if there exists a positive integer $n$ such that $D\otimes_k K\cong M(n\times n,K).$ ...
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### Complete Subalgebras and Dense Subalgebras of Complete Boolean Algebras are Regular Subalgebras

In Thomas Jech's Set Theory book, he states that If $A$ is a complete subalgebra of a complete Boolean algebra $B$ then $A$ is a regular subalgebra of $B$. Also, if $A$ is a dense subalgebra of $B$ ...
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### Finding the discriminant of a quaternion algebra

Consider the totally real number field $F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*)$. Consider the quaternion algebra $Q=(\frac{-1,-1}{F})$. How do I compute the discriminant of this algebra? I gave ...
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