Maximal orders. Reprint of the 1975 original. (English) Zbl 1024.16008

Reiner, I.

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.

Reviewer: Wolfgang Rump (Stuttgart)

MSC:

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

Keywords:

maximal orders; Dedekind domains; separable algebras; reduced norms; traces; localizations; completions; discrete valuation rings; Brauer groups; crossed products; simple algebras; hereditary orders

Citations:

Zbl 0305.16001