Some splitting theorems for algebras over commutative rings

Author:
W. C. Brown

Journal:
Trans. Amer. Math. Soc. **162** (1971), 303-315

MSC:
Primary 13.50; Secondary 16.00

DOI:
https://doi.org/10.1090/S0002-9947-1971-0284428-5

MathSciNet review:
0284428

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Abstract: Let *R* denote a commutative ring with identity and Jacobson radical *p*. Let denote the natural projection of *R* onto and a ring homomorphism such that is the identity on . 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 separable over *R*, then there exists an -algebra homomorphism such that is the identity on . Here *N* and II denote the Jacobson radical of *A* and the natural projection of *A* onto 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 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 together with the inclusion map 1 form a pair with the splitting property. Two examples are given at the end of the paper which show that being integrally closed is necessary but not sufficient to guarantee (*R, j*) has the splitting property.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0284428-5

Keywords:
Strong inertial coefficient ring,
inertial coefficient ring,
splitting property,
pair (*R, j*)

Article copyright:
© Copyright 1971
American Mathematical Society