On the axiomatic foundations of the theory of Hermitian forms

In recent work on some topological problems (7), I was forced to adopt a complicated definition of 'Hermitian form' which differed from any in the literature. A recent paper by Tits (5) on quadratic forms over division rings contains a new and simple definition of these. A major objective of this paper is to formulate both these definitions in somewhat more general terms, and to show that they are equivalent. We also discuss corresponding notions of reflexive sesquilinear forms, which also arose in topological work ((8), section 12); it is no longer true (as it is over division rings) that such forms are equivalent to hermitian or skew-symmetric ones. It is not claimed that these topics are treated below with the maximum possible generality; however, we do work with arbitrary rings (with unit), so any further generalization is likely to involve additional elements of structure (e.g. a grading or a group of operators) or a higher degree of abstraction (e.g. working over schemes instead of rings). We preface each definition by a discussion, which is intended to show some of the reasons for adopting the definition. Sesquilinear forms. Let A be a ring (with unit), M a (unital) right .4-module. We will discuss bilinear maps : M x M->A satisfying some axioms related to the module structure: these can as well be discussed for a pair M, N of right .4-modules. We think of bilinear maps of M ® N. Now the tensor product inherits any right module structure possessed by N and any left module structure possessed by M. Since A is naturally an (A-A) bimodule, it is natural to require that M be a left A -module, N a right A -module, and that J. of {A-A) bimodules. But we are only given a right A -module structure on M; this induces a left module structure over the opposite ring A. Thus to make M a left A -module, we require an isomorphism a: A -> A; which is also to be interpreted as an anti-automorphism of A. We have thus arrived at a definition. Let A be a ring with anti-automorphism a; let M, N be right A -modules. Then a map : M x N->A is oc-sesquilinear if