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Weak ideal invariance and orders in Artinian rings

Published online by Cambridge University Press:  17 April 2009

John A. Beachy
John A. Beachy
Affiliation:
Department of Mathematical Sciences, Northern Illinois Unversity, DeKalb, Illinois 60115, USA.
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Abstract

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It Is known that a ring R with Krull dimension is an order in an Artinian ring if R is K-homogeneous and the prime radical N of R is weakly ideal invariant. The notion of weak ideal invariance can be interpreted in torsion theoretic terms, yielding a shorter and more conceptual proof of this result. In addition, it is shown that the orders in Artinian rings which arise in this fashion are precisely those for which R/N is K-homogeneous.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 2 , April 1981 , pp. 255 - 264
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Beachy, John A., “Some aspects of noncommutative localization”, Noncommutative ring theory, 2–31 (Internat. Conf., Kent State University, 1975. Lecture Notes in Mathematics, 545. Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
[2]Beachy, John A., “Injective modules with both ascending and descending chain conditions on annihilators”, Comm. Algebra 6 (1978), 1777–1788.CrossRefGoogle Scholar
[3]Brown, K.A., Lenagan, T.H. and Stafford, J.T., “Weak ideal invariance and localisation”, J. London Math. Soc. (2) 21 (1980), 53–61.CrossRefGoogle Scholar
[4]Cozzens, J.H. and Sandomierski, F.L., “Localization at a semiprime ideal of a right Noetherian ring”, Comm. Algebra 5 (1977), 707–726.CrossRefGoogle Scholar
[5]Gordon, Robert, “Artinian quotient rings of FBN rings”, J. Algebra 35 (1975), 304–307.CrossRefGoogle Scholar
[6]Gordon, Robert and Robson, J.C., Krull dimension (Memoirs of the American Mathematical Society, 133. American Mathematical Society, Providence, Rhode Island, 1973).Google Scholar
[7]Jategaonkar, Arun Vinayak, “Relative Krull dimension and prime ideals in right Noetherian rings”, Comm. Algebra 2 (1974), 429–468.Google Scholar
[8]Krause, Günter, Lenagan, T.H. and Stafford, J.T., “Ideal invariance and Artinian quotient rings”, J. Algebra 55 (1978), 145–154.CrossRefGoogle Scholar
[9]Miller, Robert W. and Teply, Mark L., “The descending chain condition relative to a torsion theory”, Pacific J. Math. 83 (1979), 207–219.CrossRefGoogle Scholar
[10]Müller, Bruno J., “Ideal invariance and localization”, Comm. Algebra 7 (1979), 415–441.CrossRefGoogle Scholar
[11]Naˇstaˇsescu, Constantin, “Conditions de finitude pour les modules”, Rev. Roumaine Math. Pures Appl. 24 (1979), 745–758.Google Scholar
[12]Stenström, Bo, Rings of quotients. An introduction to methods of ring theory (Die Grundlehren der mathematischen Wissenschaften, 217. Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar