Vol. 36, No. 1, 1971

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ISSN: 0030-8730
Finite primes in simple algebras

Hoyt D. Warner

Vol. 36 (1971), No. 1, 245–265

A “prime” in an arbitrary ring with identity, as defined by D. K. Harrison, is shown to be a generalization of certain objects occurring in the classical arithmetic of a central simple K-algebra , i.e., the theory of maximal orders over Dedekind domains with quotient field K. Specifically, if K is a global field the “finite primes” of (in Harrison’s sense) which contain a K-basis for are the generators of the Erandt Groupoids of normal R-lattices, R ranging over the nontrivial valuation rings of K. The situation when contains a finite prime invariant under all K-automorphisms is studied closely; when K is the rational numbers or char (K)0, and has prime power degree, such a prime exists if and only if is a division algebra.

The techniques developed here are applied to yield new information concerning the generators and factorization in the Brandt Groupoids over certain Dedekind domains.

Mathematical Subject Classification
Primary: 16A40
Received: 11 August 1969
Published: 1 January 1971
Hoyt D. Warner