General bilinear forms

We use the generalized correspondence to show that there is a canonical set isomorphism Inn(R) \Aut⁻(R) ≅ Inn(M_n(R))\Aut⁻(M_n(R)), provided R_R 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., R_R 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.