Injective modules over a principal left and right ideal domain, with applications

I study the (indecomposable) injective modules over some noetherian rings. The main results of this dissertation take place in the setting of a principal left and right ideal domain R. One indecomposable injective is the injective envelope (divisible hull) of the module RR , and is isomorphic to the classical ring of quotients Q (or division algebra) of R. The other indecomposable injective modules are (up to isomorphism) in a one-to-one correspondence with the prime elements of the ring (up to similarity).;Motivated by a classic treatment of O.Ore [31], I take advantage of the factorization theory in R and investigate the internal structure of an indecomposable injective E ≠ Q . I describe it as a "layered" structure in two ways: first as the union of its socle series, and secondly, as the union of its elementary socle series, a concept from model theory.;More powerful results concerning these objects are obtained via the technique of localization by considering their description over an extension reals of R which is also a principal left and right ideal domain, but with a unique simple module (up to isomorphism). Each socle factor soc n(E)/socn -1(E) is a semisimple reals-module, and I show that once a choice of basis at the level soc2(E)/soc 1(E) is made, there is a canonical way to extend it through all levels so that the arithmetic of E is understandable in terms of this basis.;In addition, I analyze the right module structure of E over its endomorphism ring and study the relationship with the elementary socle series of realsE. Also, the bicommutator of realsE is shown to be the completion of the ring reals in the E-adic topology.;As a consequence of all these results, I am able to classify and describe the indecomposable injective modules over the first Weyl algebra (as well as over similar algebras), which---as wild modules---were thought to have an "unmanageable" structure [20].