0

If R is a finite-dimensional algebra over a field k, and if R is also an integral domain, show that R is a division algebra over k.

zübeyir türkoğlu
  • 31
  • 4
  • 1
    http://math.stackexchange.com/questions/63950/proof-that-an-integral-domain-that-is-a-finite-dimensional-f-vector-space-is-i – Jack Yoon Jun 17 '14 at 17:07

1 Answers1

1

If $a\in R$ then $$x\mapsto ax$$ is a linear transform, it is injective since $R$ is an integral domain. It is surjective since $R$ is finite dimensional. Therefore $1$ is in the image. Similar for right multiplication by $a$.

Rene Schipperus
  • 39,526