By Arthur Haas
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Extra resources for Atomic theory;: An elementary exposition
Example text
By lm the set of all leading coefficients of members of lnt(R) of degree m. 1. Let R be a Dedekind domain of zero characteristic and the finite norm property. lm are ideals of R. Moreover for m = 1, 2 , . . 1. 1. 2). Write r R = PI, where I is an ideal not divisible by P and observe that if c E I \ P then the polynomial g(X) - cA f . , ( X ) ~rA 32 maps R into R. 3 for every x in R the fractional ideal generated by g(x) is contained in jA and hence g(x) E R. Invoking again that corollary we get ap <_ - A .
First we define a sequence of ideals in an arbitrary domain R, which will be useful also at a later stage. Denote for m = 0, 1 , 2 , . . by lm the set of all leading coefficients of members of lnt(R) of degree m. 1. Let R be a Dedekind domain of zero characteristic and the finite norm property. lm are ideals of R. Moreover for m = 1, 2 , . . 1. 1. 2). Write r R = PI, where I is an ideal not divisible by P and observe that if c E I \ P then the polynomial g(X) - cA f . , ( X ) ~rA 32 maps R into R.
A characterization of rings R for which lnt(R) is Noetherian is not known (PROBLEM VII). 2. Let R be a domain and let A be a subring of lnt(R) containing R[X]. The ring A is called a Skolem ring (associated with R) if for every finite sequence 49 fl, 99 9 fn of polynomials belonging to A which for every r E R satisfies r~ R i=1 one has ~-~fiA = A. i=1 This condition (the Skolem property) can be also stated ill tile following equivalent way: If I is a finitely generated ideal of A such that for every a E R there is at least one polynomial f E I with f(a) = l, then I = A.