Separable Algebras Over Commutative Rings

Group Rings which are Azumaya Algebras

“…Assume FG is an Azumaya algebra and let K be the prime subfield of F. Then z(KG) is a direct summand of z(FG) as z(KG)-modules, so it follows from Lemma 2.3 and Proposition II. 1.8 of [3] that KG is an Azumaya algebra. Since Fand L have the same characteristic, K may be identified with a subfield of L. Then by applying Lemma 2.3 to K ç L we obtain that LG is an Azumaya algebra.…”

“…Our main reference for the theory of separable algebras is [3]. The reader may consult [1] for additional facts about separability.…”

“…1.8 of [3] implies that CG is an Azumaya algebra. Conversely, suppose both F and CG are Azumaya algebra.…”

“…Notice that the statement "the order of G' is invertible in F" is equivalent to the statement "no prime divisor of the order of G' is in CH(F)". 3. The finite case.…”

“…The radical, rad(FG), of FG is the ideal having the elements 1 -x as basis for x in F. By [3,Corollary II.3.7] there is an ideal / of z(FP) such that rad(FF) = IFP. We now have the vector space sums…”