Some splitting theorems for algebras over commutative rings

Abstract

Let R denote a commutative ring with identity and Jacobson radical p. Let ${\pi _0}:R \to R/p$ denote the natural projection of R onto $R/p$ and $j:R/p \to R$ a ring homomorphism such that ${\Pi _0}j$ is the identity on $R/p$. We say the pair (R, j) has the splitting property if given any R-algebra A which is faithful, connected and finitely generated as an R-module and has $A/N$ separable over R, then there exists an $(R/p)$-algebra homomorphism $j’:A/N \to A$ such that ${\Pi _{j’}}$ is the identity on $A/N$. Here N and II denote the Jacobson radical of A and the natural projection of A onto $A/N$ 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 (R, j) has the splitting property if and only if (R, j) is a strong inertial coefficient ring. If R is a Noetherian Hilbert ring with infinitely many maximal ideals such that $R/p$ 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 $(R[[x]],1)$ with the splitting property. Two examples are given at the end of the paper which show that $R/p$ being integrally closed is necessary but not sufficient to guarantee (R, j) has the splitting property.