A characterization of semilocal inertial coefficient rings

Authors:
W. C. Brown and E. C. Ingraham

Journal:
Proc. Amer. Math. Soc. **26** (1970), 10-14

MSC:
Primary 13.95

DOI:
https://doi.org/10.1090/S0002-9939-1970-0260730-2

MathSciNet review:
0260730

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Abstract: A commutative ring with identity is called an *inertial coefficient ring* if every finitely generated -algebra with separable over contains a separable subalgebra such that , where is the Jacobson radical of . Thus inertial coefficient rings are those commutative rings for which a generalization of the Wedderburn Principal Theorem holds for suitable -algebras. Our purpose is to prove that a commutative ring with only finitely many maximal ideals is an inertial coefficient ring (if and) only if it is a finite direct sum of Hensel rings.

**[1]**M. Auslander and O. Goldman,*The Brauer group of a commutative ring*, Trans. Amer. Math. Soc.**97**(1960), 367-409. MR**22**#12130. MR**0121392 (22:12130)****[2]**G. Azumaya,*On maximally central algebras*, Nagoya Math. J.**2**(1951), 119-150. MR**12**, 669. MR**0040287 (12:669g)****[3]**W. C. Brown,*Strong inertial coefficient rings*, Michigan Math. J.**17**(1970), 73-84. MR**0263802 (41:8402)****[4]**N. Bourbaki,*Algèbre commutative*. Chapitres I, II, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR**36**#146.**[5]**E. C. Ingraham,*Inertial subalgebras of algebras over commutative rings*, Trans. Amer. Math. Soc.**124**(1966), 77-93. MR**34**#209. MR**0200310 (34:209)**

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DOI:
https://doi.org/10.1090/S0002-9939-1970-0260730-2

Keywords:
Inertial coefficient ring,
Hensel ring,
inertial subalgebra,
separable algebra,
lifting idempotents

Article copyright:
© Copyright 1970
American Mathematical Society