A generalization of purely extending modules relative to a torsion theory

Abstract

In this work we introduce a new concept, namely, ${\tau }_s$-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show that for any ring $R$ with unit, if $_{R}R$ is purely ${\tau }_s$-extending if and only if every cyclic $\tau$-nonsingular $R$-module is flat. Also, we make a classification for the direct sums of the rings to be purely ${\tau }_s$-extending.

Keywords

• Pure submodule
• closed submodule
• (non)singular module
• extending module
• torsion theory

References

1. Al-Bahrani, B. H., On purely y-extending modules, Iraqi Journal of Science, 54(3) (2013), 672-675.
2. El-Bast, Z. Abd, Smith, P. F., Multiplication modules, Comm. In Algebra, 16(4) (1988), 755-779. https://doi.org/10.1080/00927878808823601
3. Anderson, F. W., Fuller, K. R., Rings and Categories of Modules. Graduate Texts in Math., No:13, Springer Verlag, New York, 1974. https://doi.org/10.1007/978-1-4612-4418-9
4. Asgari, Sh., Haghany, A., T-extending modules and t-Baer modules, Communications in Algebra, 39(5) (2011), 1605-1623. https://doi.org/10.1080/00927871003677519
5. Barnard, A., Multiplication modules, Journal of Algebra, 71 (1981), 174-178. https://doi.org/10.1016/0021-8693(81)90112-5
6. Berktaş, M. K., Dogruöz, S., Tarhan, A., Pure closed subobjects and pure quotient Goldie dimension, JP Journal of Algebra, Number Theory and Applications, 41(1) (2019), 49-57. https://doi.org/10.17654/NT041010049
7. Chatters, A. W., Hajarnavis, C. R., Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford, 28(1) (1977), 61-80. https://doi.org/10.1093/qmath/28.1.61
8. Clark, J., On purely extending modules, The Proceedings of the International Conference in Abelian Groups and Modules, (1999), 353-358. https://doi.org/10.1007/978-3-0348-7591-2-29

9. Clark, J., Lomp, C., Vanaja, N., Wisbauer, R., Lifting Modules, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006. https://doi.org/10.1007/3-7643-7573-6
10. Cohn, P. M., On the free product of associative rings, Math. Zeitchr. 71 (1959), 380-398. https://doi.org/10.1007/BF01181410
11. Crivei, S., Relatively extending modules, Algebr. Represent. Theor., 12(2-5) (2009), 319-332. https://doi.org/10.1007/s10468-009-9155-4
12. Çeken, S., Alkan, M., On τ-extending modules, Mediterranean Journal of Mathematics, 9(1) (2012), 129-142. https://doi.org/10.1007/s00009-010-0096-2
13. Doğruöz, S., Classes of extending modules associated with a torsion theory, East-West Journal of Mathematics, 8(2) (2006), 163-180.
14. Doğruöz, S., Harmancı, A., Smith, P. F., Modules with unique closure relative to a torsion theory I, Canadian Math. Bull., 53(2) (2010), 230-238. https://doi:10.3906/mat-0712-16
15. Dung, N. V., Huynh, D. V., Smith, P. F., Wisbauer, R., Extending Modules, Longman, Harlow, 1994. https://doi.org/10.1201/9780203756331
16. Ertaş, N. O., Fully Idempotent and multiplication modules, Palestine Journal of Mathematics, 3 (2014), 432-437.
17. Fieldhouse, D. J., Purity and Flatness, Ph.D. Thesis, Department of Mathematics McGill University, Montreal, Canada, July 1967.
18. Fuchs, L., Note on generalized continuous modules, preprint, (1995).
19. Golan, J. S., Torsion Theories, Longman, New York, 1986.
20. Gomez Pardo, J. L., Spectral Gabriel topologies and relative singular functors, Comm. Algebra, 13(1) (1985), 21-57. https://doi.org/10.1080/00927878508823147
21. Goodearl, K. R., Ring Theory, Nonsingular Rings and Modules, Marcel Dekker, New York, 1976.
22. Goodearl, K. R., Warfield, R. B., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 1989. https://doi.org/10.1017/CBO9780511841699
23. Harmancı, A., Smith, P. F., Finite direct sum of CS-modules, Houston J. Math., 19(4), (1993), 523-532.
24. Kamal, M. A., Muller, B. J., Extending modules over commutative domains, Osaka J. Math., 25 (1988), 531-538.
25. Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics, 189 Springer-Verlag New York, 1999. https://doi.org/10.1007/978-1-4612-0525-8
26. Rotman, J. J., An Introduction to Homological Algebra, Academic Press, New York, 1979. https://doi.org/ 10.1007/978-0-387-68324-9
27. Smith, P. F., Modules for which every submodule has a unique closure, Ring Theory Conference, World Scientific, New Jersey, (1993), 302-313.
28. Strenström, B., Rings of Quotients, Springer-Verlag, 1975.
29. Tsai, C. T., Report on Injective Modules, Queen’s Paper in Pure and Applied Mathematics, No.6, Kingston, Ontario: Queen’s University, 1965.
30. Wisbauer, R., Foundations of Module and Ring Theory, Gordon and Breach, 1991. https://doi.org/10.1201/9780203755532