A generalised character theory for modules

Crawley-Boevey's concept of a character (a notion derived from Schofield's Sylvester rank functions) is extended so that characters with different ima ge sets and classes may be analysed. A method of passing from these general ised characters to theories of modules is given and a converse technique is seen to be possible in certain cases. We consider in depth what happens wh en the character image is taken to be the ordinals with addition taken to b e the Canter sum. In particular we define a process whereby any Sigma-pure- injective module satisfying a reasonable criterion can be assigned a unique ordinal character.