The direct sum behaviour of its projective modules is a fundamental
property of any ring. Hereditary Noetherian prime rings are perhaps the
only noncommutative Noetherian rings for which this direct sum behaviour
(for both finitely and infinitely generated projective modules) is
well-understood, yet highly nontrivial.
This book surveys material previously available only in the research
literature. It provides a re-worked and simplified account, with
improved clarity, fresh insights and many original results about finite
length modules, injective modules and projective modules. It culminates
in the authors' surprisingly complete structure theorem for projective
modules which involves two independent additive invariants: genus and
Steinitz class. Several applications demonstrate its utility.
The theory, extending the well-known module theory of commutative
Dedekind domains and of hereditary orders, develops via a detailed study
of simple modules. This relies upon the substantial account of
idealizer subrings which forms the first part of the book and provides a
useful general construction tool for interesting examples.
The book assumes some knowledge of noncommutative Noetherian rings,
including Goldie's theorem. Beyond that, it is largely self-contained,
thanks to the appendix which provides succinct accounts of Artinian
serial rings and, for arbitrary rings, results about lifting direct sum
decompositions from finite length images of projective modules. The
appendix also describes some open problems.
The history of the topics is surveyed at appropriate points.
Readership
Graduate students and research mathematicians interested in
algebra, in particular, noncommutative rings.