ON ALMOST NIL-INJECTIVE RINGS

Year 2011, Volume: 9 Issue: 9, 103 - 113, 01.06.2011

Zhao Yu-e Du Xianneng

Abstract

Let R be a ring. The ring R is called right almost nil-injective, if
for any a ∈ N(R), there exists a left ideal Xa of R such that lr(a) = Ra ⊕ Xa.
In this paper, we give some characterizations and properties of almost nilinjective
rings, which is a proper generalization of AP-injective ring and almost
mininjective ring. And we study the regularity of right almost nil-injective ring,
and in the same time, when every simple singular right R−module is almost
nil-injective, we also give some properties of R .

Keywords

almost nil-injective ring , nil-injective ring , n-regular ring , almost mininjective ring

References

• F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer- Verlag, New York, 1974.
• V. Camillo, Commutative rings whose principal ideals are annihilators, Portu- gal Math., 46 (1989), 33-37.
• J.L. Chen, N.Q. Ding, Y.L. Li and Y.Q. Zhou, On (m, n)-injectivity of modules, Comm. Algebra, 29 (2001), 5589-5603.
• N.Q. Ding, M.F. Yousif and Y.Q. Zhou, Modules with annihilator conditions, Comm. Algebra, 30(5)(2002), 2309-2320.
• K.R. Goodearl, Ring Theory Nonsingular Rings and Modules, New York:Marcel Dekker, 1976.
• M.R. Goodearl, Von Neumann Regular Rings, London:Pitman, 1979.
• N.K. Kim, S.B. Nam and J.Y. Kim, On simple singular GP-injective modules, Comm. Algebra, 27(5) (1999), 2087-2096.
• S.B. Nam, N.K. Kim and J.Y. Kim, On simple GP-injective modules, Comm. Algebra, 23(14) (1995), 5437-5444.
• W.K. Nicholson, J.K. Park and M.F. Yousif, Principally quasi-injective mod- ules, Comm. Algebra, 27(4) (1999), 1683-1693.
• W.K. Nicholson and M.F. Yousif, On a theorem of Camillo, Comm.Algebra, (14) (1995), 5309-5314.
• W.K. Nicholson and M.F. Yousif, Principally injective rings, J.Algebra, 174 (1995), 77-93.
• W.K. Nicholson and M.F. Yousif, Mininjective rings, J.Algebra, 187 (1997), 578.
• W.K. Nicholson and M.F. Yousif, Weakly continuous and C2−rings, Comm. Algebra, 29(6) (2001), 2429-2446.
• S.S. Page and Y.Q. Zhou, Generalization of principally injective rings, J.Algebra, 206 (1998), 706-721.
• N.V. Sanh, K.P. Shum, S. Dhompongsa and S. Wongwai, On quasi-principally injective modules, Algebra Colloq., 6(3) (1999), 269-276.
• J.C. Wei and J.H. Chen, nil-Injective rings, Int. Electron. J. Algebra, 2 (2007), 21.
• S. Wongwai, Almost mininjective rings, Thai J.Math., 4(1) (2006), 245-249.
• R. Yue Chi Ming, On von Neumann regular rings.III, Monatsh.Math., 86(3) (1978/79), 251-257.
• Y.Q. Zhou, Rings in which certain right ideals are direct summands of annihi- lators, J. Aust. Math. Soc., 73 (2002), 335-346.
• Z.M. Zhu, J.L. Chen and X.X. Zhang, On (m, n)-quasi-injective modules, Acta Math. Univ. Comenianae, Vol.LXXIV, 1 (2005), 25-36. Zhao Yu-e
• College of Mathematics Qingdao University Qingdao, China e-mail: [e-mail] Du Xianneng
• School of Mathematics and Computational Science Anhui University Heifei, China e-mail: [e-mail]