scholarly journals Prime ideals in noncommutative valuation rings in finite dimensional central simple algebras

1993 ◽  
Vol 69 (2) ◽  
pp. 35-40 ◽  
Author(s):  
Hidetoshi Marubayashi ◽  
Haruo Miyamoto ◽  
Akira Ueda
Keyword(s):  
Prime Ideals ◽  
Central Simple Algebras ◽  
Finite Dimensional ◽  
Valuation Rings ◽  
Simple Algebras ◽  
Central Simple

UNIT GROUPS OF CENTRAL SIMPLE ALGEBRAS AND THEIR FRATTINI SUBGROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 921-932 ◽  
Author(s):  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Global Field ◽  
Central Simple Algebras ◽  
Frattini Subgroup ◽  
Division Rings ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Central Simple

Given a finite dimensional F-central simple algebra A = Mn(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that [Formula: see text] where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

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