Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T00:56:53.282Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Balcerzyk, S., On the algebraically compact groups of I. Kaplansky, Fund. Math. 44 (1957), 91–93.Google Scholar
2. Grätzer, G., Universal algebra (The University Series in Higher Mathematics, Van Nostrand, 1968).Google Scholar
3. Haley, D. K., On compact commutative Noetherian rings, Math. Ann. 189 (1970), 272–274.Google Scholar
4. Herstein, I. N., Noncommutative rings (The Carus Mathematical Monographs, John Wiley & Sons, 1968).Google Scholar
5. Kaplansky, I., Topological rings, Amer. J. Math. 69 (1947), 153–183.Google Scholar
6. Kaplansky, I., Infinite abelian groups (Univ. of Michigan Press, Ann Arbor, 1954).Google Scholar
7. Kelley, J. L., General topology (The University Series in Higher Mathematics, Van Nostrand, 1955).Google Scholar
8. Mycielski, J., Some compactifications of general algebras, Colloq. Math. 13 (1964), 1–9.Google Scholar
9. Szász, F., ÜberArtinscheRinge, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 351–354.Google Scholar
10. Warner, S., Compact Noetherian rings, Math. Ann. 141 (1960), 161–170.Google Scholar
11. Warner, S., Compact rings, Math. Ann. 145 (1962), 52–63.Google Scholar
12. Warner, S., Linearly compact Noetherian rings, Math. Ann. 178 (1968), 53–61.Google Scholar
13. Weglorz, B., Equationally compact algebras. I, Fund.Math. 59 (1966), 289–298.Google Scholar
14. Wenzel, G. H., Equational compactness in universal algebras, Manuskripte der Fakultät f. Math, und Informatik d. Univ. Mannheim, Nr. 8, 1971.Google Scholar
15. Zelinsky, D., Linearly compact modules and rings, Amer. J. Math. 75 (1953), 79–90.Google Scholar