  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 $n\in\N$), 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.
  We use the generalized correspondence to show that there is a canonical set
isomorphism between the orbits of the left action of $\Inn(R)$ on the
anti-automorphisms of $R$ and the orbits of the left action of $\Inn(M_n(R))$
on the anti-automorphisms of $M_n(R)$, provided $R_R$ is the only right
$R$-module $N$ satisfying $N^n\cong R^n$. We also 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.
