Some Splitting Theorems for Algebras Over Commutative Rings

Abstract

Abstract. Let R denote a commutative ring with identity and Jacobson radical p. Let 7t0: R -► Rjp denote the natural projection of R onto Rjp and j: Rjp ->-R a ring homomorphism such that U0j is the identity on Rjp. We say the pair (R,j) has the splitting property if given any Tî-algebra A which is faithful, connected and finitely generated as an ^-module and has AjN separable over R, then there exists an (R/p)-algebra homomorphism /': A/N -> A such that IT/' is the identity on AIN. Here N and II denote the Jacobson radical of A and the natural projection of A onto AjN respectively. The purpose of this paper is to study those pairs (R,j) which have the splitting property. If R is a local ring, then (/?,/) has the splitting property if and only if (R,j) is a strong inertial coefficient ring. If R is a Noetherian Hubert ring with infinitely many maximal ideals such that Rjp is an integrally closed domain, then (R,j) has the splitting property. If R is a dedekind domain with infinitely many maximal ideals and x an indeterminate, then the power series ring R [[x]] together with the inclusion map 1 form a pair (/? [[*]], 1) with the splitting property. Two examples are given at the end of the paper which show that Rjp being integrally closed is necessary but not sufficient to guarantee (R,j) has the splitting property.