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- ed systematically in Schilling's Book
[Sc], which are nowadays called invariant valuation rings, though
invariant valuation rings can be traced back to Hasse's work in [Has].
Since then, various attempts have been made to study the ideal theory of
orders in finite dimensional algebras over fields and to describe the
Brauer groups of fields by usage of "valuations", "places", "preplaces",
"value functions" and "pseudoplaces". In 1984, N. 1. Dubrovin defined
non-commutative valuation rings of simple Artinian rings with notion of
places in the category of simple Artinian rings and obtained significant
results on non-commutative valuation rings (named Dubrovin valuation
rings after him) which signify that these rings may be the correct def-
inition of valuation rings of simple Artinian rings. Dubrovin valuation
rings of central simple algebras over fields are, however, not
necessarily to be integral over their centers.
