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The homological dimensions of simple modules

Published online by Cambridge University Press:  17 April 2009

Nanqing Ding  and
Jianlong Chen
Nanqing Ding
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210008, Peoples Republic of China
Jianlong Chen
Affiliation:
Department of Mathematics and Mechanics, Southeast University, Nanjing 210018, Peoples Republic of China
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We prove that (a) if R is a commutative coherent ring, the weak global dimension of R equals the supremum of the flat (or (FP–)injective) dimensions of the simple R-modules; (b) if R is right semi-artinian, the weak (respectively, the right) global dimension of R equals the supremum of the flat (respectively, projective) dimensions of the simple right R-modules; (c) if R is right semi-artinian and right coherent, the weak global dimension of R equals the supremum of the FP-injective dimensions of the simple right R-modules.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 2 , October 1993 , pp. 265 - 274
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Anderson, F.W. and Fuller, K.R., Rings and categories of modules (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
[2]Auslander, M., ‘On the dimension of modules and algebras (III), global dimension’, Nagoya Math. J. 9 (1955), 67–77.Google Scholar
[3]Chase, S.U., ‘Direct products of modules’, Trans. Amer. Math. Soc. 97 (1960), 457–473.CrossRefGoogle Scholar
[4]Chen, J.L., ‘On von Neumann regular rings and SF-rings’, Math. Japon. 36 (1991), 1123–1127.Google Scholar
[5]Cozzens, J.H., ‘Homological properties of the ring of differential polynomials’, Bull. Amer. Math. Soc. 76 (1970), 75–79.Google Scholar
[6]Fieldhouse, D.J., ‘Character modules, dimension and purity’, Glasgow Math. J. 13 (1972), 144–146.Google Scholar
[7]Fields, K.L., ‘On the global dimension of residue rings’, Pacific J. Math. 32 (1970), 345–349.Google Scholar
[8]Harris, M.E., ‘Some results on coherent rings’, Proc. Amer. Math. Soc. 17 (1966), 474–479.Google Scholar
[9]Kasch, F., Modules and rings (Academic Press, London, New York, 1982).Google Scholar
[10]Osofsky, B.L., ‘Global dimension of valuation rings’, Trans. Amer. Math. Soc. 126 (1967), 136–149.Google Scholar
[11]Rainwater, J., ‘Global dimension of fully bounded Noetherian rings’, Comm. Algebra 15 (1987), 2443–2456.Google Scholar
[12]Rotman, J.J., An introduction to homological algebra (Academic Press, New York, 1979).Google Scholar
[13]Sandomierski, F.L., ‘Homological dimension under change of rings’, Math. Z. 130 (1973), 55–65.Google Scholar
[14]Smith, P.F., ‘Injective modules and prime ideals’, Comm. Algebra 9 (1981), 989–999.Google Scholar
[15]Stenström, B., ‘Coherent rings and FP-injective modules’, J. London Math. Soc. 2 (1970), 323–329.Google Scholar
[16]Stenström, B., Rings of Quotients (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[17]Teply, M.L., ‘Global dimensions of right coherent rings with left Krull dimension’, Bull. Austral. Math. Soc. 39 (1989), 215–223.Google Scholar
[18]Ware, R., ‘Endomorphism rings of projective modules’, Trans. Amer. Math. Soc. 155 (1971), 233–256.Google Scholar