Let R be a noetherian integral domain with quotient field K, and let A be a finite dimensional K-algebra. Recall that an R-order in the K-algebra A, is a sub-ring Λ of A, having the same unit element as A, and such that K . Λ = { Σαi mi (finite sum)| αi є K, mi є Λ}= A. A maximal R-order in A is defined in the usual way as such an R-order that is not properly contained in any other R-order in A. After algebraic preliminaries, basic properties of orders are investigated in Chapter 2. Maximal orders in skew-fields, over discrete valuation rings and over Dedekind domains, are studied in Chapters 3, 5 and 6, respectively. Chapter 3 is devoted to Morita equivalence and Chapter 7 deals with crossed-product algebras. The last two parts investigate simple algebras over global fields and a local and global theory of hereditary orders.

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