General bilinear forms

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 M (n) for some n a a"center dot), 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. We use the generalized correspondence to show that there is a canonical set isomorphism Inn(R) \Aut(-)(R) a parts per thousand... Inn(M (n) (R))\Aut(-)(M (n) (R)), provided R (R) is the only right R-module N satisfying N (n) a parts per thousand...R (n) , 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.