Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13.50, 16.00

Retrieve articles in all journals with MSC: 13.50, 16.00


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