Öz
A ring $R$ is called right CSP if the sum of any two closed right ideals of $R$ is also a closed right ideal of $R$. Left CSP rings can be defined similarly. An example is given to show that a left CSP ring may not be right CSP. It is shown that a matrix ring over a right CSP ring may not be right CSP. It is proved that $\mathbb{M}_{2}(R)$ is right CSP if and only if $R$ is right self-injective and von Neumann regular. The equivalent characterization is given for the trivial extension $R\propto R$ of $R$ to be right CSP.
Anahtar Kelimeler
CSP rings CS rings SSP rings self-injective rings regular rings
Destekleyen Kurum
NSFC
Proje Numarası
No.11871145 and No.12071070
Kaynakça
- [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1992.
- [2] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Longman Scientific & Technical, New York, 1994.
- [3] J. L. Garcia, Properties of direct summands of modules, Comm. Algebra 17 (1), 73-92, 1989.
- [4] K. R. Goodearl, Von Neumann Regular Rings, Monographs and Studies in Mathematics 4, Pitman, Boston, Mass.-London, 1979.
- [5] I. M-I Hadi and Th. Y. Ghawi, Modules with the closed sum property, Inter. Math. Forum 9 (32), 1539-1551, 2014.
- [6] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1998.
- [7] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, Cambridge University Press, 1990.
- [8] W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics 158, Cambridge University Press, 2003.
- [9] C. Santa-Clara, Some generalizations of injectivity, PhD thesis, University Of Glasgow, 1998.
- [10] L. Shen, A note on rings with the summand sum property, Studia Sci. Math. Hungar. 52 (4), 450-456, 2015.
- [11] L. Shen, F. Feng and W. X. Li, C-injective rings, J. Algebra Appl. 21 (12), 2250237 (13 pages), 2022.
- [12] L. Shen and W. X. Li, Generalization of CS condition, Front. Math. China 12 (1), 199-208, 2017.
Öz
Anahtar Kelimeler
CSP rings CS rings SSP rings Self-injective rings Regular rings
Proje Numarası
No.11871145 and No.12071070
Kaynakça
- [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd ed, Graduate Texts in Mathematics 13, Springer-Verlag, New York, 1992.
- [2] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Longman Scientific & Technical, New York, 1994.
- [3] J. L. Garcia, Properties of direct summands of modules, Comm. Algebra 17 (1), 73-92, 1989.
- [4] K. R. Goodearl, Von Neumann Regular Rings, Monographs and Studies in Mathematics 4, Pitman, Boston, Mass.-London, 1979.
- [5] I. M-I Hadi and Th. Y. Ghawi, Modules with the closed sum property, Inter. Math. Forum 9 (32), 1539-1551, 2014.
- [6] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1998.
- [7] S. H. Mohamed and B. J. Müller, Continuous and Discrete Modules, Cambridge University Press, 1990.
- [8] W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics 158, Cambridge University Press, 2003.
- [9] C. Santa-Clara, Some generalizations of injectivity, PhD thesis, University Of Glasgow, 1998.
- [10] L. Shen, A note on rings with the summand sum property, Studia Sci. Math. Hungar. 52 (4), 450-456, 2015.
- [11] L. Shen, F. Feng and W. X. Li, C-injective rings, J. Algebra Appl. 21 (12), 2250237 (13 pages), 2022.
- [12] L. Shen and W. X. Li, Generalization of CS condition, Front. Math. China 12 (1), 199-208, 2017.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Proje Numarası | No.11871145 and No.12071070 |
| Yayımlanma Tarihi | 15 Ağustos 2023 |
| Yayımlandığı Sayı | Yıl 2023 |