On Maximally Central Algebras

1951/02/01 by Gorô Azumaya · 241 citations
Mathematics · Chemistry · #Rings, Modules, and Algebras #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Mathematics #Center (category theory) #Modulo #Rank (graph theory) #Division algebra #Pure mathematics #Algebra over a field #Combinatorics #Algebra representation #Chemistry

paper · pdf · doi:10.1017/s0027763000010114

published in Nagoya Mathematical Journal 2, 119-150 (Cambridge University Press)

openalex publication_date 1951/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21

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.

Citations

• On the structure and ideal theory of complete local rings
• The Radical and Semi-Simplicity for Arbitrary Rings

Cited by

• Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras
• On the Theory of Henselian Rings
• Chatelet's Theorem in Synthetic Algebraic Geometry
• Henselian ring [wikipedia]
• Nakayama's lemma [wikipedia]