Noetherian ring extensions with trace conditions
Author:
Robert B. Warfield
Journal:
Trans. Amer. Math. Soc. 331 (1992), 449463
MSC:
Primary 16P40; Secondary 16D20, 16D30
MathSciNet review:
1080737
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Abstract: Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extensions without trace conditions) only under strong restrictions on the rings involved. Let be an extension of Noetherian rings such that is finitely generated as a left module and such that the left trace ideal of in is equal to . If is right fully bounded, or is a Jacobson ring, then has the same property; furthermore, and have the same classical Krull dimension. If is finitely generated as both a right and a left module, if both trace ideals of in are equal to , and if satisfies the strong second layer condition, then this condition also holds in . Finally, we compare the link graphs of and
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 K. A. Brown and R. B. Warfield, Jr., The influence of ideal structure on representation theory, J. Algebra 116 (1988), 294315. MR 953153 (89k:16026)
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 B. Cortzen and L. W. Small, Finite extensions of rings, Proc. Amer. Math. Soc. 103 (1988), 10581062. MR 954983 (89f:16020)
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 , Localization in Noetherian rings, London Math. Soc. Lecture Notes, no. 98, Cambridge Univ. Press, Cambridge, 1986. MR 839644 (88c:16005)
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 G. R. Krause and T. H. Lenagan, Growth of algebras and GelfandKirillov dimension, Pitman, London, 1985. MR 781129 (86g:16001)
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 T. H. Lenagan, Artinian ideals in Noetherian rings, Proc. Amer. Math. Soc. 51 (1975), 499500. MR 0384862 (52:5732)
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 E. S. Letzter, Primitive ideals in finite extensions of Noetherian rings, J. London Math. Soc. (2) 39 (1989), 427435. MR 1002455 (90f:16013)
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 S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Math., vol. 818, SpringerVerlag, New York, 1980. MR 590245 (81j:16041)
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 L. W. Small and R. B. Warfield, Jr., Finite extensions of rings. II, preprint, Univ. of California, San Diego.
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 L. Soueif, Normalizing extensions and injective modules, essentially bounded normalizing extensions, Comm. Algebra 15 (1987), 16071619. MR 884764 (88d:16016)
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 R. B. Warfield, Jr., Bond invariance of rings and localization, Proc. Amer. Math. Soc. 111 (1991), 1318. MR 1027102 (91d:16040)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199210807374
PII:
S 00029947(1992)10807374
Keywords:
Noetherian ring,
ring extension,
Noetherian bimodule,
trace ideal,
Jacobson ring,
Krull dimension,
fully bounded ring,
second layer condition,
link,
prime ideal
Article copyright:
© Copyright 1992 American Mathematical Society
