A generalization of purely extending modules relative to a torsion theory

Year 2021, Volume: 70 Issue: 2, 1099 - 1112, 31.12.2021

Semra Doğruöz Azime Tarhan

https://doi.org/10.31801/cfsuasmas.898637

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

Project Number

FEF-17041

References

• Al-Bahrani, B. H., On purely y-extending modules, Iraqi Journal of Science, 54(3) (2013), 672-675.
• El-Bast, Z. Abd, Smith, P. F., Multiplication modules, Comm. In Algebra, 16(4) (1988), 755-779. https://doi.org/10.1080/00927878808823601
• 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
• 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
• Barnard, A., Multiplication modules, Journal of Algebra, 71 (1981), 174-178. https://doi.org/10.1016/0021-8693(81)90112-5
• 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
• 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
• 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
• 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
• Cohn, P. M., On the free product of associative rings, Math. Zeitchr. 71 (1959), 380-398. https://doi.org/10.1007/BF01181410
• Crivei, S., Relatively extending modules, Algebr. Represent. Theor., 12(2-5) (2009), 319-332. https://doi.org/10.1007/s10468-009-9155-4
• 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
• Doğruöz, S., Classes of extending modules associated with a torsion theory, East-West Journal of Mathematics, 8(2) (2006), 163-180.
• 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
• Dung, N. V., Huynh, D. V., Smith, P. F., Wisbauer, R., Extending Modules, Longman, Harlow, 1994. https://doi.org/10.1201/9780203756331
• Ertaş, N. O., Fully Idempotent and multiplication modules, Palestine Journal of Mathematics, 3 (2014), 432-437.
• Fieldhouse, D. J., Purity and Flatness, Ph.D. Thesis, Department of Mathematics McGill University, Montreal, Canada, July 1967.
• Fuchs, L., Note on generalized continuous modules, preprint, (1995).
• Golan, J. S., Torsion Theories, Longman, New York, 1986.
• Gomez Pardo, J. L., Spectral Gabriel topologies and relative singular functors, Comm. Algebra, 13(1) (1985), 21-57. https://doi.org/10.1080/00927878508823147
• Goodearl, K. R., Ring Theory, Nonsingular Rings and Modules, Marcel Dekker, New York, 1976.
• Goodearl, K. R., Warfield, R. B., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, 1989. https://doi.org/10.1017/CBO9780511841699
• Harmancı, A., Smith, P. F., Finite direct sum of CS-modules, Houston J. Math., 19(4), (1993), 523-532.
• Kamal, M. A., Muller, B. J., Extending modules over commutative domains, Osaka J. Math., 25 (1988), 531-538.
• 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
• Rotman, J. J., An Introduction to Homological Algebra, Academic Press, New York, 1979. https://doi.org/ 10.1007/978-0-387-68324-9
• Smith, P. F., Modules for which every submodule has a unique closure, Ring Theory Conference, World Scientific, New Jersey, (1993), 302-313.
• Strenström, B., Rings of Quotients, Springer-Verlag, 1975.
• Tsai, C. T., Report on Injective Modules, Queen’s Paper in Pure and Applied Mathematics, No.6, Kingston, Ontario: Queen’s University, 1965.
• Wisbauer, R., Foundations of Module and Ring Theory, Gordon and Breach, 1991. https://doi.org/10.1201/9780203755532