1

Let $\operatorname{End}_K(V)$ be the endomorphism ring of the K-vector space V. Are there any non-trivial central idempotent elements of $\operatorname{End}_K(V)$? I already know that if $f\in \operatorname{End}_K(V)$ and $f$ is idempotent then $V=\operatorname{Ker}(f)\oplus \operatorname{Im}(f)$. Is this fact useful?

epimorphic
  • 3,270
Dismal
  • 149

1 Answers1

1

The center of $\operatorname{End}_K(V)$ is the set of uniform scaling maps $\lambda \mathrm{Id}$. If such an element is also idempotent, you have $\lambda(\lambda-1)\mathrm{Id}=0$. Hence the identity and the zero linear map are the only possible central idempotent elements.

epimorphic
  • 3,270
mathcounterexamples.net
  • 71,758
  • Explain to me why f is in the center implies that it is like you suggested. – Dismal Jan 09 '16 at 15:55
  • You can have a look at http://math.stackexchange.com/questions/299626/the-center-of-operatornamegln-k which provides a proof oro the finite dimensional case that can be adapted to the general case. – mathcounterexamples.net Jan 09 '16 at 16:06