We use the generalized correspondence to show that there is a canonical set isomorphism Inn(R) \Aut−(R) ≅ Inn(Mn(R))\Aut−(Mn(R)), provided RR is the only right R-module N satisfying Nn ≅Rn, and also to prove a variant of a theorem of Osborn. Namely, we classify all semisimple rings with involution admitting no non-trivial idempotents that are invariant under the involution.
We introduce the new notion of general bilinear forms (generalizing sesquilinear forms) and prove that for every ring R (not necessarily commutative, possibly without involution) and every right R-module M which is a generator (i.e., RR is a summand of Mn for some n ∈ ℕ), there is a one-to-one correspondence between the anti-automorphisms of End(M) and the general regular bilinear forms on M, considered up to similarity. This generalizes a well-known similar correspondence in the case R is a field. We also demonstrate that there is no such correspondence for arbitrary R-modules.
|Number of pages||39|
|Journal||Israel Journal of Mathematics|
|State||Published - Feb 2015|
Bibliographical notePublisher Copyright:
© 2015, Hebrew University of Jerusalem.
ASJC Scopus subject areas
- Mathematics (all)