Hereditary Noetherian Prime Rings 1. Integrality and Simple Modules

This is the first of three papers that aim to bring the known theory of projective modules over a hereditary Noetherian prime ring R up to roughly the same level as the well-known commutative case, where R is a Dedekind domain. This first paper lays the foundations by introducing the notion of an integral extension S of R in the Goldie quotient ring of R, and elucidating the relationship between integrality and the R-module structure of simple S-modules.