Pairs of Inverse Modules in a Skewfield

Let S be a skewfield. If J and / ' are submodules of 2 such that the nonzero elements of J are the inverse elements of those of J , then J and J' form a "pair of inverse modules." A module admitting an inverse module will be called a /-module and a selfinverse module containing 1 will be called an 5-module. In an earlier paper the author has shown that if S is a (commutative) field of characteristic not equal to 2, then every 5-module is a subfield of S. Only in fields of characteristic 2, nontrivial 5-modules can be found. A corresponding distinction of that characteristic does not hold for skewfields. Even the skewfield of the quaternions contains nontrivial 5-modules, for examples the module generated by 1, J, k. In the present paper some properties of 5-modules and /-modules will be discussed. For example it will be proved that when an 5-module contains the elements a, b and aby it contains all the elements of the skewfield which is generated by a and b. By a similar method it will be shown that finite 5-modules are necessarily Galois-fields.