Separable algebras over commutative rings

Some Splitting Theorems for Algebras Over Commutative Rings

“…S being separable over R implies S is central separable over CiS), the center of S, and CiS) is separable over R. Thus CiS) is a commutative, finitely generated, faithful, connected, and separable /î-algebra. Using [9,Corollary 4.2] again, we get C(S) is projective over R. Now by Lemma 1, parts (c) and (d), of this paper, the Jacobson radical, rad iCiS)), of CiS) is generated by the radical of R which is 0. Therefore rad (C(iS")) = 0.…”

“…Thus C(A/N), the center of A/N, is a finitely generated, faithful, connected, separable and commutative Ä-algebra. Since R is integrally closed, it now follows from [9,Corollary 4.2] that C(A/N) is projective over R. A/N itself is projective over C(A/N). Thus A/N is projective over R.…”

“…Now A/N is a finitely generated, faithful, connected and separable .R-algebra. Thus using [9,Corollary 4.2] In this section, we give two examples concerning Theorem 3. One may first ask if the hypothesis that R be integrally closed is necessary.…”

“…The discriminant of x2 + x-1 is 5. Whence it follows from [9,Theorem 2.2] that A is not a separable Ä-algebra, but A/mA is separable over R/m for all maximal ideals m of R except for that maximal ideal M of R lying over the ideal (5) = -2yx + y3 -5y4 = 0, yf in Zs}.…”

On the Brauer splitting theorem

“…41-1, p. 292 and Corollary 70-24, p. 475). In [8], G. J. Janusz defined a ring R to be a splitting ring for G if the group algebra RG is the direct sum of central separable iϋ-algebras each equivalent to R in the Brauer group of R; that is, RG ^ 0 ΣU Hom^ (P if P^), where {PJ are finitely generated projective faithful j?-modules, the number of different conjugate classes in G is equal to s. He then proved the Brauer splitting theorem for a Noetherian regular domain, R. This section gives a proof for the above theorem when R is any commutative ring with no idempotents except 0 and 1.…”