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- W2153669848 abstract "An example is given of an uncountable Noetherian ring whose additive group is This answers a question posed indirectly by L. Fuchs. In [2] on page 311 read: we do not know of any uncountable Noetherian ring whose additive group is free. Our purpose is to give an example of such a ring. Let A be the commutative associative polynomial ring Z [x, . ,xi . .. ., i C I, where Z is the ring of integers, each xi is indeterminate, and I is the set of positive ordinals less than ol, the first uncountable ordinal. Let S be the set consisting of those members of A which are relatively prime to each positive integer. Let R be the subring of the field Z (xI, ... , xi, ... ) generated by those members which can be written in the form p/q with p c A and q C S. We will prove the following theorem. THEOREM 1. The ring R is an uncountable Noetherian ring with free additive group. In order to prove this theorem, shall need the following theorem which was proved in a more general form by Paul Hill (see Theorem 3 of [3]). THEOREM A. Let 01 be the first uncountable ordinal. If F1 5 F2 5 * C Fj C.. ., j< 1, is an ascending chain of free subgroups of an abelian group G, indexed by the positive ordinals less than (1, such that (i) Fk = Uj<kFj if k is a limit ordinal less than 'l, (ii) G = Uj<,w,Fjg (iii) I Fj < K for each j, then G is free if Fj+ 1/Fj is free for each positive i < ?1. In this paragraph explain some terminology. Whenever an element in R is expressed in rational form p/q, it is assumed that p is in A and q is in S. Let x0 = 1, the identity element in R. Forj C I, let (Fj, + ) be the subgroup of (R, + ) generated by those members which can be written in the formp/q where p and q are polynomials in {xi: 0 < i <K). If j E I andp E Fj+i n A, by degj p mean the degree of p considered as a polynomial in xj over Received by the editors March 9, 1977. AMS (MOS) subject classifications (1970). Primary 20K20, 16A46." @default.
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- W2153669848 date "1977-02-01" @default.
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- W2153669848 title "An uncountable Noetherian ring with free additive group" @default.
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- W2153669848 doi "https://doi.org/10.1090/s0002-9939-1977-0450426-2" @default.
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