# Questions tagged [algebras]

For questions about algebras, their properties, and their structures.

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### Modules over the dual of an infinite dimensional coalgebra

Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of ...
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### Well-defined way to quotient by a relation involving an infinite sum

Take a unital (*-)algebra $\mathcal A$ generated by a finite set of generators, $e_n$, and relations. We can require the generators to be (self-adjoint) projections. Some of the relations are of the ...
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### A quotient of tensor algebra of $V \otimes k$

Let $k$ be a field and $V$ a $k$-vector space. Denote the inclusion of $k$ in $k \oplus V$ by $m$. A book that I'm consulting (Ramanan's Global Calculus) asks the reader to prove that the quotient of ...
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Let $D$ be a composition algebra over a field $F$. Define on $C = D \oplus D$ a product by, $$(x,y)(u,v)=(xu + \lambda \bar{v}y, vx+y \bar{u})$$ and a quadratic form $N:C \to F$ by $$N((x,y))= N(x) -... 1answer 66 views ### Right adjoint of a monad is a comonad I'm having trouble proving the following statement: If a monad T has a right adjoint K, then K is a comonad and the categories of T-algebras and K-coalgebras are isomorphic. So far I've ... 0answers 27 views ### Difference between functors \text{Hom}_B(-,B) and \text{Hom}_{\mathbb C}(-,\mathbb C) I am working with a commutative finite-dimensional \mathbb C-algebra B and I am supposed to consider and prove a theorem about the hom-functors$$\text{Hom}_B(-,B)\quad\text{and}\quad \text{Hom}_{\...
I wanted to ask if someone had a proof for the following claim: Given and integral domain $A$ and a finitely generated $A$-algebra $B$, show that there exists elements $x_1,...,x_n \in B$ ...