A GENERALIZATION OF SIMPLE-INJECTIVE RINGS

Abstract

A ring R is called right 2-simple J-injective if, for every 2-generated

right ideal I < J(R), every R-linear map from I to R with simple image extends to R. The class of right 2-simple J-injective rings is broader than that of

right 2-simple injective rings and right simple J-injective rings. Right 2-simple

J-injective right Kasch rings are studied, several conditions under which right

2-simple J-injective rings are QF-rings are given.

Keywords

• 2-Simple J-injective ring,Kasch ring,semilocal ring,semiperfect ring,QF-ring

References

1. J. E. Bjork, Rings satisfying certain chain conditions, J. Reine Angew. Math., 245 (1970), 63-73.
2. J. L. Chen, N. Q. Ding and M. F. Yousif, On Noetherian rings with essential socle, J. Aust. Math. Soc., 76 (2004), 39-49.
3. J. L. Chen, Y. Q. Zhou and Z. M. Zhu, GP-injective rings need not be P-injective, Comm. Algebra, 33 (2005), 2395-2402.
4. J. L. Gomez Pardo and P. A. Guil Asensio, Torsionless modules and rings with nite essential socle, Abelian groups, module theory, and topology (Padua, 1997), Lecture Notes in Pure and Appl. Math., Dekker, New York, 201 (1998), 261-278.
5. M. Harada, Self mini-injective rings, Osaka J. Math. 19 (1982), 587-597.
6. L. X. Mao, Min-flat modules and min-coherent rings, Comm. Algebra, 35 (2007), 635-650.
7. W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra, 174 (1995), 77-93.
8. W. K. Nicholson and M. F. Yousif, Mininjective rings, J. Algebra, 187 (1997), 548-578.

9. W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics, 158, Cambridge University Press, Cambridge, 2003.
10. W. K. Nicholson, J. K. Park and M. F. Yousif, Extensions of simple-injective rings, Comm. Algebra, 28 (2000), 4665-4675.
11. W. K. Nicholson, J. K. Park and M. F. Yousif, On simple-injective rings, Algebra Colloq., 9 (2002), 259-264.
12. S. S. Page and Y. Q. Zhou, Generalizations of principally injective rings, J. Algebra, 206 (1998), 706-721.
13. E. A. Rutter, Jr., Rings with the principal extension property, Comm. Algebra, 3 (1975), 203-212.
14. L. Shen and J. L. Chen, New characterizations of quasi-Frobenius rings, Comm. Algebra, 34 (2006), 2157-2165.
15. M. F. Yousif and Y. Q. Zhou, Rings for which certain elements have the principal extension property, Algebra Colloq., 10 (2003), 501-512.
16. M. F. Yousif and Y. Q. Zhou, FP-injective, simple-injective, and quasi-Frobenius rings, Comm. Algebra, 32 (2004), 2273-2285.
17. Y. Q. Zhou, Rings in which certain right ideals are direct summands of annihilators, J. Aust. Math. Soc., 73 (2002), 335-346.
18. Z. M. Zhu and J. L. Chen, 2-Simple injective rings, Int. J. Algebra, 4 (2010), 25-37.
19. Z. M. Zhu, MP-injective rings and MGP-injective rings, Indian J. Pure. Appl. Math., 41 (2010), 627-645.
20. Z. M. Zhu, Some results on MP-injectivity and MGP-injectivity of rings and modules, Ukrainian Math. J., 63 (2012), 1623-1632.