Much progress has been made during the last decade on the subjects of non- commutative valuation rings, and of semi-hereditary and Priifer orders in a simple Artinian ring which are considered, in a sense, as global theories of non-commu- tative valuation rings. So it is worth to present a survey of the subjects in a self-contained way, which is the purpose of this book. Historically non-commutative valuation rings of division rings were first treat-...