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.
