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Goldie dimensions of quotient modules

Part of: Modules, bimodules and ideals Chain conditions, growth conditions, and other forms of finiteness

Published online by Cambridge University Press:  09 April 2009

John Dauns
John Dauns
Affiliation:
Department of Mathematics Tulane UniversityNew Orleans, LA 70118USA e-mail: jd@math.tulane.edu
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Abstract

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For an infinite cardinal ℵ an associative ring R is quotient ℵ<-dimensional if the generalized Goldie dimension of all right quotient modules of RR are strictly less than ℵ. This latter quotient property of RR is characterized in terms of certain essential submodules of cyclic modules being generated by less than ℵ elements, and also in terms of weak injectivity and tightness properties of certain subdirect products of injective modules. The above is the higher cardinal analogue of the known theory in the finite ℵ = ℵ0 case.

Keywords

quotient finite dimensionalweakly injectivecomplement submodule

MSC classification

Secondary: 16D50: Injective modules, self-injective rings 16P60: Chain conditions on annihilators and summands
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 1 , August 2001 , pp. 11 - 19
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Al-Huzali, A. H., Jain, S. K. and López-Permouth, S. R., ‘Rings whose cyclics have finite Goldie dimension’, J. Algebra 128 (1990), 257–269.Google Scholar
[2]Al-Huzali, A. H., Jain, S. K. and López-Permouth, S. R., ‘On the weak relative injectivity of rings and modules’, in: Noncommutative ring theory (eds. Jain, S. K. and Lopéz, S. R.), Lecture Notes in Math. 1448 (Springer, New York, 1990) pp. 93–98.CrossRefGoogle Scholar
[3]Camillo, V. P., ‘Modules whose quotients have finite Goldie dimension’, Pacific. J. Math. 69 (1977), 337–338.CrossRefGoogle Scholar
[4]Dauns, J., ‘Subdirect products of injecitves’, Comm. Algebra 17 (1989), 179–196.CrossRefGoogle Scholar
[5]Dauns, J., ‘Generalized finiteness conditions’, Comm. Algebra 22 (1994), 2877–2895.CrossRefGoogle Scholar
[6]Dauns, J., ‘Module types’, Rocky Mountain J. Math. 27 (1997), 503–557.CrossRefGoogle Scholar
[7]Dauns, J. and Fuchs, L., ‘Infinite Goldie dimensions’, J. Algebra 115 (1988), 297–302.CrossRefGoogle Scholar
[8]Gordon, R. and Robson, J. C., ‘Krull dimension’, Memoirs of the Amer. Math. Soc. 133 (1973), 1–78.Google Scholar
[9]Jain, S. K. and López-Permouth, S. R., ‘A survey on the theory of weakly injective modules’, in: Computational algebra (eds. Fisher, K. G., Loustaunau, P., Shapiro, J., Green, E. L. and Farkas, D.), Lecture Notes in Pure and Appl. Math. 151 (Marcel Dekker, New York, 1994) pp. 205–232.Google Scholar
[10]Kurshan, R. P., ‘Rings whose cyclics have finitely generated socle’, J. Algebra 15 (1970), 376–386.CrossRefGoogle Scholar
[11]López-Permouth, S. R., ‘Rings characterized by their weakly-injective modules’, Glasgow Math. J. 34 (1992), 349–353.CrossRefGoogle Scholar