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., 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.