Finite primes in simple algebras

Abstract

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 i-algebra 2, 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 2 (in Harrison's sense) which contain a i-basis for 2 are the generators of the Brandt Groupoids of normal ^-lattices, R ranging over the nontrivial valuation rings of K. The situation when J contains a finite prime invariant under all i-automorphisms is studied closely; when K is the rational numbers or char (IT) 0, and 2 has prime power degree, such a prime exists if and only if J is a division algebra.