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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Rings which are almost polynomial rings


Authors: Paul Eakin and James Silver
Journal: Trans. Amer. Math. Soc. 174 (1972), 425-449
MSC: Primary 13F20
MathSciNet review: 0309924
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Abstract: If A is a commutative ring with identity and B is a unitary A-algebra, B is locally polynomial over A provided that for every prime p of A, $ {B_p} = B{ \otimes _A}{A_p}$ is a polynomial ring over $ {A_p}$. For example, the ring $ Z[\{ X/{p_i}\} _{i = 1}^\infty ]$, where $ \{ {p_i}\} _{i = 1}^\infty $ is the set of all primes of Z, is locally polynomial over Z, but is not a polynomial ring over Z. If B is locally polynomial over A, the following results are obtained, B is faithfully flat over A. If A is an integral domain, so is B. If $ \mathfrak{a}$ is any ideal of A, then $ B/\mathfrak{a}B$ is locally polynomial over $ A/\mathfrak{a}$. If p is any prime of A, then pB is a prime of B. If B is a Krull ring, so is A and the class group of B is isomorphic to the class group of A . If A is a Krull ring and B is contained in an affine domain over A, then B is a Krull ring. If A is a noetherian normal domain and B is contained in an affine ring over A, then B is a normal affine ring over A. If M is a module over a ring A, the content of an element x of M over A is defined to be the smallest ideal $ {A_x}$ of A such that x is in $ {A_x}M$. A module is said to be a content module over A if $ {A_x}$ exists for every x in M. M is a content module over A if and only if arbitrary intersections of ideals of A extend to M. Projective modules are content modules. If B is locally polynomial over a Dedekind domain A, then B is a content module over A if and only if B is Krull.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0309924-4
PII: S 0002-9947(1972)0309924-4
Article copyright: © Copyright 1972 American Mathematical Society