5 - Modules over firs and semifirs

Summary

Just as firs form a natural generalization of principal ideal domains, so there is a class of modules over firs that generalizes the finitely generated modules over principal ideal domains. They are the positive modules studied in Section 5.3; they admit a decomposition into indecomposables, with a Krull–Schmidt theorem (in fact this holds quite generally for finitely presented modules over firs), but it is no longer true that the indecomposables are cyclic. On the other hand, there is a dual class, the negative modules, and we shall see how the general finitely presented module is built up from free modules, positive and negative modules. A basic notion is that of a bound module; this and the duality, essentially the transpose, also used in the representation theory of algebras, are developed in Sections 5.1 and 5.2 in the more general context of hereditary rings. In the special case of free algebras, the endomorphism rings of finitely presented bound modules are shown to be finite-dimensional over the ground field. This result, first proved by J. Lewin, is obtained here by means of a normal form for matrices over a free algebra, due to M. L. Roberts, and his work is described in Section 5.8.

A second topic is the rank of matrices. Several notions of rank are defined, of which the most important, the inner rank, is studied more closely in Section 5.4. Over a semifir the inner rank obeys Sylvester's law of nullity.