Pairs of Inverse Moduls

Two submodules A and A' of a (commutative) field F will be said to be inverse, if to every element a≠0 of A, there exists in A' the inverse element a' = a^-1, and conversely. If A' is the same module as A, then A is said to be self-inverse. The rational homogeneous functions of x, y, z of order m with coefficients from an arbitrary field K, e.g. form a module which is inverse to a module formed by homogeneous functions of order - m. If m≠0, the two moduls have no common element, besides zero; if m = 0, the module is a field and therefore self-inverse.