General Bilinear Forms

Original · EN

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∈), there is a one-to-one correspondence between the anti-automorphisms of (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 (R) on the anti-automorphisms of R and the orbits of the left action of (Mₙ(R)) on the anti-automorphisms of Mₙ(R), provided Rᵣ is the only right R-module N satisfying Nⁿ Rⁿ. 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.