Research Article
Year 2020,
Volume: 49 Issue: 3, 914 - 920, 02.06.2020
Abstract
References
- [1] E. Büyükaşık and Y. Durğun, Neat-flat Modules, Comm. Algebra, 44 (1), 416–428, 2016.
- [2] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhäuser Verlag, Basel, 2006.
- [3] J. Clark and P.F. Smith, On semi-Artinian modules and injectivity conditions, Proc. Edinburgh Math. Soc. 39 (2), 263–270, 1996.
- [4] S. Crivei and S. Şahinkaya, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math. 18 (4), 989–1002, 2014.
- [5] S.E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121, 223–235, 1966.
- [6] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules. Pitman Research Notes in Math. Ser. 313, Longman Scientific & Technical, Harlow, 1994.
- [7] Y. Durğun, On some generalizations of closed submodules, Bull. Korean Math. Soc. 52 (5), 1549–1557, 2015.
- [8] Y. Durğun and S. Özdemir, On S-closed submodules, J. Korean Math. Soc. 54 (4), 1281–1299, 2017.
- [9] L. Fuchs, Neat submodules over integral domains, Period. Math. Hungar. 64 (2), 131–143, 2012.
- [10] K.R. Goodearl, Singular torsion and the splitting properties. Amer. Math. Soc. 124, Providence, R. I., 1972.
- [11] K.R. Goodearl, Ring theory, Marcel Dekker, Inc., New York-Basel, 1976.
- [12] K.R. Goodearl, von Neumann regular rings, Monographs and Studies in Mathematics, 4, Pitman Boston, Mass, 1979.
- [13] A. Harmanci and P.F. Smith, Finite direct sums of CS-modules, Houston J. Math. 19 (4), 523–532, 1993.
- [14] Y. Kara and A. Tercan, When some complement of a z-closed submodule is a sum- mand, Comm. Algebra, 46 (7), 3071–3078, 2018.
- [15] T.Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, Springer- Verlag, New York, 1999.
- [16] K. Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (3), 310–338, 1984.
- [17] K. Oshiro, On Harada rings. I, II, Math. J. Okayama Univ. 31, 161–178, 179-188, 1989.
- [18] J.J. Rotman, An introduction to homological algebra., Second, Universitext, Springer, New York, 2009.
- [19] F.L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19, 225–230, 1968.
- [20] A. Tercan, On CLS-modules Rocky Mountain J. Math. 25 (4), 1557–1564, 1995.
- [21] J. Wang and D. Wu, When an S-closed submodule is a direct summand, Bull. Korean Math. Soc. 51 (3), 613–619, 2014.
- [22] H. Zöschinger, Schwach-Flache Moduln, Comm. Algebra, 41 (12), 4393–4407, 2013.
Year 2020,
Volume: 49 Issue: 3, 914 - 920, 02.06.2020
Abstract
A submodule $N$ of a module $M$ is called d-closed if $M/N$ has a zero socle. D-closed submodules are similar concept to s-closed submodules, which are defined through nonsingular modules by Goodearl. In this article we deal with modules with the property that all d-closed submodules are direct summands (respectively, closed, pure). The structure of a ring over which d-closed submodules of every module are direct summand (respectively, closed, pure) is studied.
References
- [1] E. Büyükaşık and Y. Durğun, Neat-flat Modules, Comm. Algebra, 44 (1), 416–428, 2016.
- [2] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhäuser Verlag, Basel, 2006.
- [3] J. Clark and P.F. Smith, On semi-Artinian modules and injectivity conditions, Proc. Edinburgh Math. Soc. 39 (2), 263–270, 1996.
- [4] S. Crivei and S. Şahinkaya, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math. 18 (4), 989–1002, 2014.
- [5] S.E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121, 223–235, 1966.
- [6] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules. Pitman Research Notes in Math. Ser. 313, Longman Scientific & Technical, Harlow, 1994.
- [7] Y. Durğun, On some generalizations of closed submodules, Bull. Korean Math. Soc. 52 (5), 1549–1557, 2015.
- [8] Y. Durğun and S. Özdemir, On S-closed submodules, J. Korean Math. Soc. 54 (4), 1281–1299, 2017.
- [9] L. Fuchs, Neat submodules over integral domains, Period. Math. Hungar. 64 (2), 131–143, 2012.
- [10] K.R. Goodearl, Singular torsion and the splitting properties. Amer. Math. Soc. 124, Providence, R. I., 1972.
- [11] K.R. Goodearl, Ring theory, Marcel Dekker, Inc., New York-Basel, 1976.
- [12] K.R. Goodearl, von Neumann regular rings, Monographs and Studies in Mathematics, 4, Pitman Boston, Mass, 1979.
- [13] A. Harmanci and P.F. Smith, Finite direct sums of CS-modules, Houston J. Math. 19 (4), 523–532, 1993.
- [14] Y. Kara and A. Tercan, When some complement of a z-closed submodule is a sum- mand, Comm. Algebra, 46 (7), 3071–3078, 2018.
- [15] T.Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, Springer- Verlag, New York, 1999.
- [16] K. Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (3), 310–338, 1984.
- [17] K. Oshiro, On Harada rings. I, II, Math. J. Okayama Univ. 31, 161–178, 179-188, 1989.
- [18] J.J. Rotman, An introduction to homological algebra., Second, Universitext, Springer, New York, 2009.
- [19] F.L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19, 225–230, 1968.
- [20] A. Tercan, On CLS-modules Rocky Mountain J. Math. 25 (4), 1557–1564, 1995.
- [21] J. Wang and D. Wu, When an S-closed submodule is a direct summand, Bull. Korean Math. Soc. 51 (3), 613–619, 2014.
- [22] H. Zöschinger, Schwach-Flache Moduln, Comm. Algebra, 41 (12), 4393–4407, 2013.
There are 22 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | June 2, 2020 |
| Published in Issue | Year 2020 Volume: 49 Issue: 3 |