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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The preparation theorem and the freeness of $A[[X]]/I$


Author: S. H. Cox
Journal: Proc. Amer. Math. Soc. 61 (1976), 227-231
MSC: Primary 14D15; Secondary 13B05, 13J05, 14B10
DOI: https://doi.org/10.1090/S0002-9939-1976-0427317-5
MathSciNet review: 0427317
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Abstract: Let $I$ be a nonzero ideal of $A[[X]]$, the ring of formal power series over a commutative Noetherian ring $A$. These are equivalent: (i) $I$ is generated by a single series $f = {a_0} + {a_1}X + \ldots$ such that for some $s,\;{a_s}$ is a unit, the first $s$ coefficients ${a_0}, \ldots ,{a_{s - 1}}$ of $f$ lie in the Jacobson radical of $A$ and $A$ is complete in the adic topology defined by the ideal they generate. (ii) $A[[X]]/I$ is a free $A$-module.


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Keywords: Formal power series ring, free module, projective module, analytic ring extension, Preparation Theorem
Article copyright: © Copyright 1976 American Mathematical Society