A. W. Chatters [72] has proved a decomposition theorem for a hereditary, Noetherian ring R, namely[image omitted] ,where Ri is either a prime ring or an Artinian ring[image omitted] . An immediate consequence is a theorem of Small [66] stating that R has a classical right2 quotient ring Q(R), since Q(R) = R for any Artinian R, and Q(R) exists for any prime Noetherian ring R by the theorems of Goldie [58, 60] and Lesieur-Croisot [59]. Unfortunately, Chatters' proof uses a lemma from Small's paper, marring the stated implication. In this paper, we give a another proof of Chatters' theorem, not only eliminating the dependency on the lemma of Small, but also generalizing it, and a number of other lemmas, from which we deduce a theorem of Camillo and Cozzens [73], characterizing when a left Ore domain is a principal left ideal ring assuming it is a principal right ideal domain.
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