# Weakly np-Injective Rings and Weakly C2 Rings

Wei, Junchao;Che, Jianhua

• Accepted : 2010.12.27
• Published : 2011.03.31
• 11 4

#### Abstract

A ring R is called left weakly np- injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R- homomorphism from $Ra^n$ to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np- injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and $_RR$ is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np- injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n- regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.

#### Keywords

Left weakly np- injective rings;Left weakly C2 rings;Directly finite rings;Regular rings

#### References

1. R. Camps and W. Dicks, On semi-local rings, Israel J. Math., 81(1993), 203-211. https://doi.org/10.1007/BF02761306
2. J. L. Chen and N. Q. Ding, On General principally injective rings, Comm. Alg., 27(1999), 2097-2116. https://doi.org/10.1080/00927879908826552
3. J. L. Chen and N. Q. Ding, On regularity of rings, Alg. Colloq., 8(3)(2001), 267-274.
4. C. Faith and P. Menal, A counter example to a conjecture of Johns, Proc. Amer. Math. Soc., 116(1992), 21-26. https://doi.org/10.1090/S0002-9939-1992-1100651-0
5. C. Faith and P. Menal, The structure of Johns rings, Proc. Amer. Math. Soc., 120(1994), 1071-1081. https://doi.org/10.1090/S0002-9939-1994-1231294-8
6. Y. C. R. Ming, On Quasi-Frobeniusean and Artinian rings, Publications De L, institut Math ematique, 33(47)(1983), 239-245.
7. W. K. Nicholson and E. S. Campos, Rings with the dual of the isomorphism theorem, J. Alg., 271(2004), 391-406. https://doi.org/10.1016/j.jalgebra.2002.10.001
8. W. K. Nicholson and J. F. Watters, Rings with projective socle, Proc. Amer. Math. Soc., 102(1988), 443-450. https://doi.org/10.1090/S0002-9939-1988-0928957-5
9. W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Alg., 174(1995), 77-93. https://doi.org/10.1006/jabr.1995.1117
10. W. K. Nicholson and M. F. Yousif, Minijective rings, J. Alg., 187(1997), 548-578. https://doi.org/10.1006/jabr.1996.6796
11. W. K. Nicholson and M. F. Yousif, Annihilators and the CS-condition, Glasgow Math. J., 40(1998), 213-222. https://doi.org/10.1017/S0017089500032535
12. W. K. Nicholson and Yousif M. F., On finitely embedded rings, Comm. Alg., 28(11)(2000), 5311-5315. https://doi.org/10.1080/00927870008827157
13. W. K. Nicholson and M. F. Yousif, Weakly continuous and C2-rings, Comm. Alg., 29(6)(2001), 2429-2466. https://doi.org/10.1081/AGB-100002399
14. J. L. G. Pardo and P. A. G. Asensio, Rings with nite essential socle, Proc. Amer. Math. Soc., 125(1997), 971-977. https://doi.org/10.1090/S0002-9939-97-03747-7
15. S. Page and Y. Q. Zhou, Generalizations of principally injective rings, J. Alg., 206(3)(1998), 706-721. https://doi.org/10.1006/jabr.1998.7403
16. J. C. Wei and J. H. Chen, Nil- injective rings, Intern. Electr. J. Alg., 3(2007), 1-21.
17. J. C. Wei and J. H. Chen, NPP rings, reduced rings and SNF rings, Intern. Electr. J. Alg., 4(2008), 9-26.
18. Y. Q. Zhou, Rings in which certain right ideals are direct summands of annihilators, J. Aust. Math. Soc., 73(2002), 335-346. https://doi.org/10.1017/S1446788700009009