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Maximal orders. Reprint of the 1975 original. (English) Zbl 1024.16008
London Mathematical Society Monographs. New Series. 28. Oxford: Oxford University Press. xiv, 395 p. (2003).
28 years after its first appearence, Irving Reiner’s book “Maximal orders” has remained to be a standard reference to “non-commutative arithmetics”. Apart from the classical applications already mentioned in the book, there are some areas of currently active research, like non-commutative algebraic geometry or non-Abelian Iwasawa theory, where the theory of maximal orders has become an important requisite.
Not only for maximal orders, but also for general orders over a Dedekind domain, Reiner’s book provides an excellent introduction for students and serves as an indispensible reference for researchers. For a summary of contents we refer to the review Zbl 0305.16001 of the original.

16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16U30 Divisibility, noncommutative UFDs
16K20 Finite-dimensional division rings
16P50 Localization and associative Noetherian rings
11R52 Quaternion and other division algebras: arithmetic, zeta functions
16G30 Representations of orders, lattices, algebras over commutative rings
11S45 Algebras and orders, and their zeta functions
14F22 Brauer groups of schemes
01A75 Collected or selected works; reprintings or translations of classics