Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-20T23:35:25.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Subdirectly irreducible rings–some pathology

Published online by Cambridge University Press:  17 April 2009

H.G. Moore
H.G. Moore
Affiliation:
Brigham Young University, Provo, Utah.
Rights & Permissions [Opens in a new window]

Abstract

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.

Every ring is isomorphic to a subdirect sum of subdirectly irreducible rings. Unfortunately, however, as is shown, the property of being subdirectly irreducible is not preserved under homomorphisms. An example is given of a finite non-commutative subdirectly irreducible ring R with heart (= the intersection of all non-zero ideals) H, such that R/E is isomorphic with GF(2) + GF(2). (GF(2) is the two element Galois Field.) Some additional properties of the ring R are listed and contrasts are made with results for commutative subdirectly irreducible rings; for example, the zero divisors of R do not form an ideal.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 3 , December 1969 , pp. 353 - 355
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Divinsky, Nathan, Rings and radicals (University of Toronto Press, Toronto, 1965).Google Scholar
[2]Divinsky, Nathan, “Commutative subdirectly irreducible rings”, Proc. Amer. Math. Soc. 8 (1957), 642–648.CrossRefGoogle Scholar
‘3’Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
[4]McCoy, Neal H., “Subdirectly irreducible commutative rings”, Duke Math. J. 12 (1945), 381–387.CrossRefGoogle Scholar
[5]Moore, H.G. and Yaqub, Adil, “A generalization of boolean rings”, to appear.Google Scholar
[6]Pierce, Richard S., Introduction to the theory of abstract algebras (Holt, Rinehart & Winston, New York, 1968).Google Scholar