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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

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