The global dimension of FBN rings with enough clans

Author:
Robert F. Damiano

Journal:
Proc. Amer. Math. Soc. **86** (1982), 25-28

MSC:
Primary 16A60; Secondary 16A33

DOI:
https://doi.org/10.1090/S0002-9939-1982-0663859-8

MathSciNet review:
663859

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Abstract: For an FBN ring , a classical set of prime ideals is one for which the semiprime ideal satisfies the Artin-Rees property. A minimal classical set is called a clan. We say an FBN ring has enough clans if each prime ideal is an element of a clan. In this paper, we show that for such rings the Krull dimension is less than or equal to the global dimension.

**[1]**F. W. Anderson and K. R. Fuller,*Rings and categories of modules*, Graduate Texts in Math., Springer-Verlag, Berlin and New York, 1974. MR**0417223 (54:5281)****[2]**H. Bass,*On the ubiquity of Gorenstein rings*, Math. Z.**82**(1963), 8-28. MR**0153708 (27:3669)****[3]**M. Boratynski,*A change of rings theorem and the Artin-Rees property*, Proc. Amer. Math. Soc.**53**(1975), 307-310. MR**0401840 (53:5667)****[4]**R. Damiano and Z. Papp,*On consequences of stability*, Comm. Algebra**9**(1981), 747-764. MR**609221 (82h:16032)****[5]**-,*Stable rings with finite global dimension*, Hudson Symposium on Recent Advances in Non-Commutative Ring Theory at Plattsburgh, New York, Springer-Verlag, Berlin and New York.**[6]**A. V. Jategaonkar,*Injective modules and localization in noncommutative noetherian rings*, Trans. Amer. Math. Soc.**190**(1974), 109-123. MR**0349727 (50:2220)****[7]**-,*Relative Krull dimension and prime ideals in right noetherian rings*, Comm. Algebra (**2**) (1974), 429-468. MR**0357494 (50:9962)****[8]**S. Jondrup,*Homological dimensions of some P.I. rings*, Comm. Algebra**8**(1980), 685-696. MR**561548 (82f:16013)****[9]**B. Mueller,*Localization in fully bounded noetherian rings*, Pacific J. Math.**67**(1976), 233-245. MR**0427351 (55:385)****[10]**-,*Localization of non-commutative noetherian rings at semiprime ideals*, Lecture notes, McMaster Univ., 1975.**[11]**R. Resco, L. Small and J. T. Stafford,*Krull and global dimensions of semiprime noetherian**-rings*(preprint).**[12]**J. J. Rotman,*An introduction to homological algebra*, Academic Press, New York, 1980. MR**538169 (80k:18001)****[13]**J. Shapiro,*-sequences in fully bounded noetherian rings*Comm. Algebra**7**(1979), 819-831. MR**529495 (81c:16004)****[14]**B. Stenström,*Rings of quotients*, Springer-Verlag, Berlin and New York, 1975. MR**0389953 (52:10782)****[15]**D. Ž. Djoković,*Epimorphisms of modules which must be isomorphisms*, Canad. Math. Bull.**16**(1973), 513-515. MR**0346014 (49:10740)**

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0663859-8

Article copyright:
© Copyright 1982
American Mathematical Society