Rings with Minimum Condition

Commutative Banach Algebras with Idempotent Maximal Ideals

“…PROOF. If 33 is a radical algebra the result is trivially true 3 . Otherwise, let 33" be the algebra 33 with an identity e adjoined if necessary; by Lemma 3 we may suppose that e e 91.…”

“…But this means that 93, and hence 21, is finite dimensional. That 21 has an identity follows from [3] Corollary 4.3B.…”

Equivalence Classes of Maximal Orders

“…Consider the map φ: H\G ι G 2 ,U{L 1 L 2 )) -> H^GyG^Z*) defined at the beginning of this section. Since (0([flrj)) n i = [1] and (φilgΫW* = [1] it is clear that {φ{[g 1 gl 1 ])) n^ = [1] so that the Brauer number of 2 divides n x n 2 and is hence relatively prime to the characteristic of R.…”

“…), so that β{l) = 0 . Therefore flf(<y, p) is a p t/ι root of unity since for some p th root of unity ζ tf since a is in G Next let a denote an element of GJG 1 and p any element of G w . Then g(σ 9 p) = Π e^) /<j(Π e^> ).…”

“…Consider a 2-cocycle g z in Z 2 (G I9 U(L)) which is cohomologous to f x and which is normalized in the sense of cyclic groups. Let φ 1 Let L 7 denote the fixed field of G 7 and let a denote the element of U{Lj) which defines the 2-cocycle g x . Since L is a tamely ramified inertial extension of L I9 the natural map # 2 (G 7 , U(S))->H 2 {G I9 U{L)) is an epimorphism, where 5 denotes the integral closure of R in L (see the proof of Cor.…”