Reiner, I. 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. Reviewer: Wolfgang Rump (Stuttgart) Cited in 126 Documents 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 PDF BibTeX XML Cite \textit{I. Reiner}, Maximal orders. Reprint of the 1975 original. Oxford: Oxford University Press (2003; Zbl 1024.16008) OpenURL