Hereditary Noetherian Prime Rings 2. Finitely Generated Projective Modules

Abstract

Let R be a hereditary Noetherian prime ring. We determine a full set of invariants for the isomorphism class of any finitely generated projective R-module of uniform dimension at least 2. In particular we prove that P ⊕ X ∼ = Q ⊕ X implies P ∼ = Q whenever P has uniform dimension at least 2. Among the applications of these results are necessary and sufficient conditions for the existence of a bound to the number of generators needed for right ideals of R.

Hereditary Noetherian prime rings, 3: Infinitely generated projective modules

“…As with Kaplansky's result, this displays a simplification when compared with the corresponding result for finitely generated projective modules [12,Theorem 4.4] where the Steinitz class of P and P must also match. One consequence is the determination, in terms of genera, of when one projective R-module P is isomorphic to a proper direct summand of a given infinitely generated projective R-module Q.…”

“…There is a relatively complete structure theory for finitely generated projective R-modules which appears in [12]. That, in turn, relies on results from [11] about simple R-modules and their extensions.…”

“…Our recent paper [12] describes the structure of finitely generated projective modules over R. Here we complete the analysis of projective modules over these rings (which, in the commutative case, become the Dedekind domains studied by Kaplansky).…”

Hereditary Noetherian Prime Rings 1. Integrality and Simple Modules

“…The theory developed in this section enables us to describe precisely those integral overrings S of R which fall into these two special classes. Such overrings are used in the companion article [8] to this paper. Proof.…”

“…This programme is completed in two companion papers [8,9]. Here we present the theory of integral overrings independently of its connection with projective R-modules, mainly for clarity of exposition but also because it is of interest in its own right.…”