Homologically homogeneous rings

Brown, K. A.; Hajarnavis, C. R.

doi:10.1090/S0002-9947-1984-0719665-5

Homologically homogeneous rings
HTML articles powered by AMS MathViewer

by K. A. Brown and C. R. Hajarnavis PDF

Trans. Amer. Math. Soc. 281 (1984), 197-208 Request permission

Abstract:

In this paper we study the structure of a right Noetherian ring $R$ of finite right global dimesion integral over a central subring $C$ and satisfying the following condition: if $V,W$ are irreducible right $R$-modules with ${r_C}(V) = {r_C}(W)$ then $\operatorname {pr} \dim (V) = \operatorname {pr} \dim (W)$.

References

G. M. Bergman and P. M. Cohn, The centres of $2$-firs and hereditary rings, Proc. London Math. Soc. (3) 23 (1971), 83–98. MR 291200, DOI 10.1112/plms/s3-23.1.83
Joseph Bernstein, Israël Gel′fand, and Serge Gel′fand, Structure locale de la catégorie des modules de Harish-Chandra, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 11, A495–A497 (French, with English summary). MR 489155
S. M. Bhatwadekar, On the global dimension of some filtered algebras, J. London Math. Soc. (2) 13 (1976), no. 2, 239–248. MR 404398, DOI 10.1112/jlms/s2-13.2.239
K. A. Brown, C. R. Hajarnavis, and A. B. MacEacharn, Noetherian rings of finite global dimension, Proc. London Math. Soc. (3) 44 (1982), no. 2, 349–371. MR 647437, DOI 10.1112/plms/s3-44.2.349
K. A. Brown, C. R. Hajarnavis, and A. B. MacEacharn, Rings of finite global dimension integral over their centres, Comm. Algebra 11 (1983), no. 1, 67–93. MR 687406, DOI 10.1080/00927878308822837
Marc Chamarie, Localisations dans les ordres maximaux, Comm. Algebra 2 (1974), 279–293 (French). MR 352154, DOI 10.1080/00927877408822013
Marc Chamarie and Alain Hudry, Anneaux noethériens à droite entiers sur un sous-anneau de leur centre, Comm. Algebra 6 (1978), no. 2, 203–222 (French). MR 463241, DOI 10.1080/00927877808822242
A. W. Chatters and C. R. Hajarnavis, Rings with chain conditions, Research Notes in Mathematics, vol. 44, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR 590045
Robert Gordon and J. C. Robson, Krull dimension, Memoirs of the American Mathematical Society, No. 133, American Mathematical Society, Providence, R.I., 1973. MR 0352177
Andy J. Gray, A class of maximal orders integral over their centres, Glasgow Math. J. 24 (1983), no. 2, 177–180. MR 706147, DOI 10.1017/S0017089500005255
S. Jøndrup, Homological dimensions of some P.I. rings, Comm. Algebra 8 (1980), no. 7, 685–696. MR 561548, DOI 10.1080/00927878008822483
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 0470211
Richard Resco, Lance W. Small, and J. T. Stafford, Krull and global dimensions of semiprime Noetherian PI-rings, Trans. Amer. Math. Soc. 274 (1982), no. 1, 285–295. MR 670932, DOI 10.1090/S0002-9947-1982-0670932-1
J. C. Robson, Some constructions of rings of finite global dimension, Glasgow Math. J. 26 (1985), no. 1, 1–11. MR 776670, DOI 10.1017/S001708950000570X
Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
Bo Stenström, Rings of quotients, Die Grundlehren der mathematischen Wissenschaften, Band 217, Springer-Verlag, New York-Heidelberg, 1975. An introduction to methods of ring theory. MR 0389953
Richard G. Swan, Groups of cohomological dimension one, J. Algebra 12 (1969), 585–610. MR 240177, DOI 10.1016/0021-8693(69)90030-1
Richard B. Tarsy, Global dimension of orders, Trans. Amer. Math. Soc. 151 (1970), 335–340. MR 268226, DOI 10.1090/S0002-9947-1970-0268226-3
Wolmer V. Vasconcelos, On quasi-local regular algebras, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 11–22. MR 0330159