Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-15T10:16:21.557Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on Noetherian orders in Artinian rings

Published online by Cambridge University Press:  18 May 2009

A. W. Chatters
A. W. Chatters
Affiliation:
School of Mathematics, University Walk, Bristol BS8 1TW
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.

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 20 , Issue 2 , July 1979 , pp. 125 - 128
Copyright
Copyright © Glasgow Mathematical Journal Trust 1979

References

REFERENCES

1.Chatters, A. W., Two results on p.p. rings, Comm. Algebra 4 (1976), 881–891.CrossRefGoogle Scholar
2.Ginn, S. M. and Moss, P. B., A decomposition theorem for Noetherian orders in Artinian rings, Bull. London Math. Soc. 9 (1977), 177–181.CrossRefGoogle Scholar
3.Lenagan, T. H., Artinian ideals in Noetherian rings, Proc. Amer. Math. Soc. 51 (1975), 499–500.Google Scholar
4.Robson, J. C., Decomposition of Noetherian rings, Comm. Algebra 1 (1974), 345–349.CrossRefGoogle Scholar
5.Small, L. W., Orders in Artinian rings, J. Algebra 4 (1966), 13–41.CrossRefGoogle Scholar
6.Small, L. W., Orders in Artinian rings II, J. Algebra 9 (1968), 266–273.CrossRefGoogle Scholar
7.Small, L. W., Hereditary rings, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 25–27.CrossRefGoogle ScholarPubMed
8.Small, L. W., Semi-hereditary rings, Bull. Amer. Math. Soc. 73 (1967), 656–658.CrossRefGoogle Scholar
9.Small, L. W., On some questions in Noetherian rings, Bull. Amer. Math. Soc. 72 (1966), 853–857.CrossRefGoogle Scholar
10.Small, L. W., An example in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1035–1036.CrossRefGoogle ScholarPubMed