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Some Splitting Theorems for Algebras Over Commutative Rings

1971; American Mathematical Society; Volume: 162; Linguagem: Inglês

10.2307/1995755

1088-6850

W. C. Brown,

Magnolia and Illicium research

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.Introduction.In [3], the notion of a strong inertial coefficient ring iR,j) was first introduced.Let R be a commutative ring with identity.Let p denote the Jacobson radical of R and Il0: /?-> Rjp the natural projection of R onto R/p.We assume there exists a ring homomorphism j: R/p -> R, mapping R/p into R, such that Tl0j is the identity map of R/p.Then the pair (R,j) is called a strong inertial coefficient ring if the following property is satisfied : Given any /{-algebra A which is finitely generated as an /î-module and has A/N separable over R, then there exists an (/?//?)-algebra homomorphism j' : A/N -> A such that Uj' is the identity on A/N.Here II and N denote the natural projection of A onto A/N and the Jacobson radical of A respectively.In [3], we showed that if (R,j) was a strong inertial coefficient ring, then R itself was an inertial coefficient ring.In [4], the author and E. Ingraham determined the structure of all semilocal inertial coefficient rings.Namely, R is an inertial coefficient ring with finitely many maximal ideals if and only if R is a finite direct sum of Hensel rings.Thus we can characterize semilocal strong inertial coefficient rings as follows: The pair (R,j) is a strong inertial coefficient ring if and only if R is a finite direct sum of split Hensel rings.