Research Article
Year 2021,
Volume: 50 Issue: 2, 516 - 525, 11.04.2021
Abstract
References
- [1] I. Amin, Y. Ibrahim and M .F. Yousif, $C3$-modules, Algebra Colloq. 22 (4), 655–670, 2015.
- [2] I. Amin, M.F. Yousif and N. Zeyada, Soc-injective rings and modules, Commun. Algebra, 33, 4229–4250, 2005.
- [3] F.W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
- [4] K.I. Beidar and W. F. Ke, On essential extensions of direct sums of injective modules, Archiv. Math. 78, 120–123, 2002.
- [5] V. Camillo, Y. Ibrahim, M. Yousif and Y. Zhou, Simple-direct-injective modules, J. Algebra 420, 39–53, 2014.
- [6] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Math. 313, Longman, Harlow, New York, 1994.
- [7] E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39, 189–209, 1981.
- [8] J.W. Fisher, Von Neumann regular rings versus V-rings, in: Lect. Notes Pure Appl. Math. 7, 101–119, Dekker, New York, 1974.
- [9] K.R. Goodearl, Von Neumann Regular Rings, Krieger Publishing Company, Malabar, Florida, 1991.
- [10] J. Hausen, Modules with the summand intersection property, Commun. Alg. 17, 135– 148, 1989.
- [11] Y. Ibrahim, M.T. Koşan, M. Yousif and T.C. Quynh, Simple-direct-projective modules, Commun. Algebra, 44 (12), 5163–5178, 2014.
- [12] Y. Ibrahim, T.C. Quynh and M. Yousif, Simple-direct-modules, Commun. Algebra, 45 (8), 3643–3652, 2017.
- [13] S.H. Mohammed and B.J. Müller, Continous and Discrete Modules, London Math. Soc. LN 147, Cambridge Univ. Press., 1990.
- [14] W.K. Nicholson and M.F. Yousif, Quasi-Frobenius Rings, Cambridge Univ. Press., 2003.
- [15] T.C. Quynh and M.T. Koşan, On annihilators and quasi-Frobenius rings, submitted.
- [16] S. Sahinkaya and J. Trlifaj, Generalized injectivity and approximations, Commun. Algebra, 44 (9), 4047–4055, 2014.
- [17] G.V. Wilson, Modules with the Direct Summand Intersection Property, Comm. Alg. 14, 21–38, 1986.
- [18] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
Year 2021,
Volume: 50 Issue: 2, 516 - 525, 11.04.2021
Abstract
The notion of simple-direct-injective modules which are a generalization of injective modules unifies $C2$ and $C3$-modules. In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of $C2$, $C3$, SSP properties and simple-direct-injective modules. It is proved that a ring $R$ is Artinian serial with the Jacobson radical square zero if and only if every semisimple-direct-injective right $R$-module has the SSP and, for any family of simple injective right $R$-modules $\{S_i\}_{\mathcal{I}}$, $\oplus_{\mathcal{I}}S_i$ is injective. We also show that $R$ is a right Noetherian right V-ring if and only if every right $R$-module has a semisimple-direct-injective envelope if and only if every right $R$-module has a semisimple-direct-injective cover.
References
- [1] I. Amin, Y. Ibrahim and M .F. Yousif, $C3$-modules, Algebra Colloq. 22 (4), 655–670, 2015.
- [2] I. Amin, M.F. Yousif and N. Zeyada, Soc-injective rings and modules, Commun. Algebra, 33, 4229–4250, 2005.
- [3] F.W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
- [4] K.I. Beidar and W. F. Ke, On essential extensions of direct sums of injective modules, Archiv. Math. 78, 120–123, 2002.
- [5] V. Camillo, Y. Ibrahim, M. Yousif and Y. Zhou, Simple-direct-injective modules, J. Algebra 420, 39–53, 2014.
- [6] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Math. 313, Longman, Harlow, New York, 1994.
- [7] E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39, 189–209, 1981.
- [8] J.W. Fisher, Von Neumann regular rings versus V-rings, in: Lect. Notes Pure Appl. Math. 7, 101–119, Dekker, New York, 1974.
- [9] K.R. Goodearl, Von Neumann Regular Rings, Krieger Publishing Company, Malabar, Florida, 1991.
- [10] J. Hausen, Modules with the summand intersection property, Commun. Alg. 17, 135– 148, 1989.
- [11] Y. Ibrahim, M.T. Koşan, M. Yousif and T.C. Quynh, Simple-direct-projective modules, Commun. Algebra, 44 (12), 5163–5178, 2014.
- [12] Y. Ibrahim, T.C. Quynh and M. Yousif, Simple-direct-modules, Commun. Algebra, 45 (8), 3643–3652, 2017.
- [13] S.H. Mohammed and B.J. Müller, Continous and Discrete Modules, London Math. Soc. LN 147, Cambridge Univ. Press., 1990.
- [14] W.K. Nicholson and M.F. Yousif, Quasi-Frobenius Rings, Cambridge Univ. Press., 2003.
- [15] T.C. Quynh and M.T. Koşan, On annihilators and quasi-Frobenius rings, submitted.
- [16] S. Sahinkaya and J. Trlifaj, Generalized injectivity and approximations, Commun. Algebra, 44 (9), 4047–4055, 2014.
- [17] G.V. Wilson, Modules with the Direct Summand Intersection Property, Comm. Alg. 14, 21–38, 1986.
- [18] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
There are 18 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | April 11, 2021 |
| Published in Issue | Year 2021 Volume: 50 Issue: 2 |