General Bilinear Forms

TL;DR

This paper introduces a broad concept of bilinear forms for rings and modules, establishing a correspondence with anti-automorphisms, and explores their properties and classifications, extending classical results from fields to more general rings.

Contribution

It generalizes the correspondence between anti-automorphisms and bilinear forms from fields to arbitrary rings and modules, and classifies certain involutions on semisimple rings.

Findings

Established a one-to-one correspondence for generators over rings

Demonstrated the non-existence of such correspondence for arbitrary modules

Classified semisimple rings with involution without non-trivial invariant idempotents

Abstract

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.