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 image sets and classes may be analysed. A method of passing from these generalised 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 when the character image is taken to be the ordinals with addition taken to be the Cantor sum. In particular we define a process whereby any Σ-pure-injective module satisfying a reasonable criterion can be assigned a unique ordinal character.