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Splitting torsion theories over commutative rings

Published online by Cambridge University Press:  17 April 2009

John D. Fuelberth ,
James Kuzmanovich  and
Thomas S. Shores
John D. Fuelberth
Affiliation:
Department of Mathematics, University of Northern Colorado, Greeley, Colorado, USA;
James Kuzmanovich
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, Nebraska, USA.
Thomas S. Shores
Affiliation:
Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina, USA;
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Abstract

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The purpose of this paper is to completely characterize splitting torsion theories over commutative rings. In particular, if (T, F) is a torsion theory for which T(R) = 0, then (T, F) is a splitting theory if and only if T contains only a finite number of nonisomorphic simple modules and every module in T is semisimple injective. In addition, an ideal theoretic characterization of splitting torsion theories is given, of which one consequence is that splitting torsion theories are TTF; furthermore, if R is also noetherian, then such torsion theories are centrally splitting. The known theorems concerning the splitting of the Goldie and simple torsion theories (for commutative rings) are derived from our theorem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 3 , December 1975 , pp. 457 - 464
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Cateforis, Vasily C. and Sandomierski, Francis L., “The singular submodule splits off”, J. Algebra 10 (1968), 149–165.Google Scholar
[2]Goodearl, K.R., Singular torsion and the splitting properties (Mem. Amer. Math. Soc. 124. Amer. Math. Soc., Providence, Rhode Island Island, 1972).Google Scholar
[3]Henriksen, M. and Jerison, M., “The space of minimal prime ideals of a commutative ring”, Trans. Amer. Math. Soc. 115 (1965), 110–130.Google Scholar
[4]Levine, Jeffrey, “On the injective hulls of semisimple modules”, Trans. Amer. Math. Soc. 155 (1971), 115–126.CrossRefGoogle Scholar
[5]Rosenberg, Alex and Zelinsky, Daniel, “Finiteness of the injective hull”, Math. Z. 70 (1958/1959), 372–380.CrossRefGoogle Scholar
[6]Rotman, Joseph, “A characterization of fields among the integral domains”, An. Acad. Brasil. Ci. 32 (1960), 193–194.Google Scholar
[7]Sarath, B. and Varadarajan, K., “Injectivity of direct sums”, Comm. Algebra 1 (1974), 517–530.CrossRefGoogle Scholar
[8]Teply, Mark L., “Some aspects of Goldie's torsion theory”, Pacific J. Math. 29 (1969), 447–459.Google Scholar
[9]Teply, Mark L. and Fuelberth, John D., “The torsion submodule splits off”, Math. Ann. 188 (1970), 270–284.Google Scholar