SOME STUDIES ON GZI RINGS

Abstract

A ring R is called generalized ZI (or GZI for short) if for any

a 2 N(R) and b 2 R, ab = 0 implies aRba = 0, which is a proper generalization

of ZI rings. In this paper, many properties of GZI rings are introduced, some

known results are extended. Further, we introduce generalized GZI rings

as a generalization of GZI rings, and quasi-abel rings as a generalization of

generalized GZI rings. Some important results on Abel rings are extended to

generalized GZI rings and quasi-abel rings.

Keywords

• ZI ring,GZI ring,quasi-abel ring,generalized GZI ring,reduced ring

References

1. A. Badawi, On abelian pi-regular rings, Comm. Algebra, 25(4) (1997), 1009- 1021.
2. M. Baser, A. Harmanci and T. K. Kwak, Generalized semicommutative rings and their extensions, Bull. Korean Math. Soc., 45(2) (2008), 285-297.
3. M. Baser and T. K. Kwak, Extended semicommutative rings, Algebra Colloq., 17(2) (2010), 257-264.
4. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc., 2 (1970), 363-368.
5. A. Y. M. Chin, Clean elements in abelian rings, Proc. Indian Acad. Sci. Math. Sci., 119(2) (2009), 145-148.
6. P. M. Cohn, Reversible rings, Bull. London Math. Soc., 31(6) (1999), 641-648. [7] G. Ehrlich, Unit-regular rings, Portugal. Math., 27 (1968), 209-212.
7. M. Henriksen, Two classes of rings generated by their units, J. Algebra, 31 (1974), 182-193.
8. S. U. Hwang, Y. C. Jeon and K. S. Park, On NCI rings, Bull. Korean Math. Soc., 44(2) (2007), 215-223.

9. N. K. Kim, S. B. Nam and J. Y. Kim, On simple singular GP-injective modules, Comm. Algebra, 27(5) (1999), 2087-2096.
10. J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull., 14 (1971), 359-368.
11. R. Mohammadi, A. Moussavi and M. Zahiri, On nil-semicommutative rings, Int. Electron. J. Algebra, 11 (2012), 20-37.
12. W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
13. W. K. Nicholson and M. F. Yousif, Mininjective rings, J. Algebra, 187(2) (1997), 548-578.
14. G. Y. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc., 184 (1973), 43-60.
15. L. N. Vaserstein, Bass's rst stable range condition, J. Pure Appl. Algebra, 34(2-3) (1984), 319-330.
16. J. C. Wei, On simple singular Y J-injective modules, Southeast Asian Bull. Math., 31(5) (2007), 1009-1018.
17. J. C. Wei, Certain rings whose simple singular modules are nil-injective, Tur- kish J. Math., 32(4) (2008), 393-408.
18. J. C. Wei and J. H. Chen, Nil-injective rings, Int. Electron. J. Algebra, 2 (2007), 1-21.
19. J. C. Wei and L. B. Li, Strong DS rings, Southeast Asian Bull. Math., 33(2) (2009), 375-390.
20. J. C. Wei and L. B. Li, Quasi-normal rings, Comm. Algebra, 38(5) (2010), 1855-1868. J. C. Wei and Y. C. Qu, On rings containing a non-essential nil-injective maximal left ideal, Kyungpook Math. J., 52(2) (2012), 179-188.
21. T. S. Wu and P. Chen, On nitely generated projective modules and exchange rings, Algebra Colloq., 9(4) (2002), 433-444.
22. H.-P. Yu, On quasi-duo rings, Glasgow Math. J., 37(1) (1995), 21-31.
23. H.-P. Yu, Stable range one for exchange rings, J. Pure Appl. Algebra, 98(1) (1995), 105-109.