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Equationally Compact Artinian Rings

Published online by Cambridge University Press:  20 November 2018

David K. Haley
David K. Haley*
Affiliation:
Universität Mannheim, Mannheim, West Germany
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By a Noetherian (Artinian) ring = (R; + , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ring is said to be equationally compact if every system of ring polynomial equations with constants in is simultaneously solvable in provided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 2 , April 1973 , pp. 273 - 283
Copyright
Copyright © Canadian Mathematical Society 1973

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