On Maximally Central Algebras

Abstract

Let A be a primary algebra with unit element over a field K and Z its center. Let Ā be the simple residue class algebra of A modulo its radical. Then it is known, and can readily be seen, that there holds the inequality where t is the rank of A over its center. We call A maximally central if in particular i.e. if the rank [Z: K] takes its maximum value. Further, an algebra which is a direct sum of those primary algebras will be called maximally central too. The notion was introduced in Azumaya-Nakayama [5], as a by-product of the study of absolutely uni-serial algebras.