Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T15:00:27.315Z Has data issue: false hasContentIssue false

The Torsion Submodule of A Cyclic Module Splits Off

Published online by Cambridge University Press:  20 November 2018

Mark L. Teply*
Affiliation:
University of Florida, Gainesville, Florida
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A prominent question in the study of modules over an integral domain has been: “When is the torsion submodule t(A) of a module A a direct summand of A?” A module is said to split when its torsion module is a direct summand. Clearly, every cyclic module over an integral domain splits. Interesting splitting problems have been explored by Kaplansky [14; 15], Rotman [20], Chase [4], and others.

Recently, many concepts of torsion have been proposed for modules over arbitrary associative rings with identity. Two of the most important of these concepts are Goldie's torsion theory (see [1; 12; 22]) and the simple torsion theory (see [5; 6; 8; 9; 23], and their references).

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Alin, J. S. and Dickson, S. E., Goldie's torsion theory and its derived functor, Pacific J. Math. 24 (1968), 195203.Google Scholar
2. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, 1956).Google Scholar
3. Cateforis, V. and Sandomierski, F., The singular submodule splits off, J. Algebra 10 (1968), 149165.Google Scholar
4. Chase, S. U., Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457473.Google Scholar
5. Dickson, S. E., A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223235.Google Scholar
6. Dickson, S. E., Noetherian splitting rings are artinian, J. London Math. Soc. J+2 (1967), 732-736.Google Scholar
7. Endo, S., On semi-hereditary rings, J. Math. Soc. Japan 13 (1961), 109119.Google Scholar
8. Fuchs, L., Torsion preradicals and ascending Lowey series of modules, J. Reine Angew. Math. 239-240 (1969), 169179.Google Scholar
9. Fuelberth, J. D., On commutative splitting rings, Proc. Amer. Math. Soc. 20 (1970), 393408.Google Scholar
10. Gabriel, P., Des catégories abélienes, Bull. Math. Soc. France 90 (1962), 323448.Google Scholar
11. Gilmer, R., Multiplicative ideal theory (Queen's University Papers on Mathematics, No. 12, Kingston, 1968).Google Scholar
12. Goldie, A. W., Torsionfree rings and modules, J. Algebra 1 (1964), 268287.Google Scholar
13. Jans, J. P., Some aspects of torsion, Pacific J. Math. 15 (1965), 12491259.Google Scholar
14. Kaplansky, I., Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327340.Google Scholar
15. Kaplansky, I., A characterization of Prüfer rings, J. Indian Math. Soc. 24 (1960), 279281.Google Scholar
16. Maranda, J. M., Infective structures, Trans. Amer. Math. Soc. 110 (1964), 98135.Google Scholar
17. Matlis, E., Modules with descending chain condition, Trans. Amer. Math. Soc. 97 (1960), 495508.Google Scholar
18. Năstăsescu, C. and Popescu, N., Anneaux semi-artiniens, Bull. Soc. Math. France 96 (1968), 357368.Google Scholar
19. Renault, G., Etude des sons-modules dans un module, Bull. Soc. Math. France Mem. 9 (1967).Google Scholar
20. Rotman, J., A characterization of fields among the integral domains, Anais. Acad. Brasil Ciêna 32 (1960), 193194.Google Scholar
21. Smith, J. R., Local domains with topologically T-nilpotent radical, Pacific J. Math. 30 (1969), 233245.Google Scholar
22. Teply, M. L., Some aspects of Goldie's torsion theory, Pacific J. Math. 29 (1969), 447459.Google Scholar
23. Teply, M. L. and Fuelberth, J. D., The torsion submodule splits off, Math. Ann. 188 (1970), 270284.Google Scholar
24. Zariski, O. and Samuel, P., Commutative algebra. I. (D. van Nostrand Company, Princeton, 1958).Google Scholar