2

Skolem-Noether Theorem: Let S be a finite dimensional central simple k-algebra, and let R be a simple k-algebra. If f,g: R-> S are homomorphism (necessarily one-to-one), then there is an inner automorphism T:S -> S such that Tf=g.

What happens if we replace the central simple algebra S in the Skolem-Noether Theorem with a semisimple ring? Show that with the appropriate centralizer assumption, the first part of the theorem will still hold, but the second part will not. What happens if one drops the assumption that the ring homomorphisms take 1 to 1?

Could you help me about these questions especially for the last one? Generally, we are interested with the ring homomorphisms take 1 to 1. But now i am a bit confused.

Çiğdem
  • 21
  • 1
  • I notice in the statement that you mention the map is one-to-one and then in the last paragraph you talk about taking 1 to 1. You're aware that "one-to-one" is a synonym for "injective," and is not really referring to preservation of the identities, right? – rschwieb Jun 16 '14 at 16:43

1 Answers1

0

Yeah you're right. I think i misunderstood this question. I know that one to one is a synonym for injective but i thought that means referring to preservation of identities. But nevertheless I have problems with these questions.

Çiğdem
  • 1