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Direct Sums of Torsion-Free Covers

Published online by Cambridge University Press:  20 November 2018

Thomas Cheatham
Thomas Cheatham*
Affiliation:
Samford University, Birmingham, Alabama
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In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 5 , October 1973 , pp. 1002 - 1005
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Banaschewski, B., On coverings of modules, Math. Nachr. 31 (1966), 57–71.Google Scholar
2. Bass, H., Global dimension of rings, Ph.D. Thesis, University of Chicago, 1959.Google Scholar
3. Cheatham, T., Finite dimensional torsion-free rings, Pacific J. Math. 39 (1971), 1–6.Google Scholar
4. Enochs, E., Torsion-free covering modules. II, Arch. Math. (Basel) 22 (1971), 37–52.Google Scholar
5. Goldie, A. W., Torsion-free modules and rings, J. Algebra 1 (1964), 268–287.Google Scholar
6. Rentschler, R., Die vertauschbarkeit des Hom-functor mit direkten summen, Docktoral These, University Munchen, Munich, 1967.Google Scholar
7. Rentschler, R., Sur les modules M tels que Horn (M, —) commute avec les sommes directes, C. R. Acad. Sci. Paris Sér. A-B 288 (1969), 930–933.Google Scholar
8. Sandomierski, F. L., Semi-simple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120.Google Scholar
9. Teply, M., Torsion-free infective modules, Pacific J. Math. 28 (1969), 441–453.Google Scholar