##
**AGQP-injective modules.**
*(English)*
Zbl 1160.16002

Summary: Let \(R\) be a ring and let \(M\) be a right \(R\)-module with \(S=\text{End}(M_R)\). \(M\) is called ‘almost general quasi-principally injective’ (or AGQP-injective for short) if, for any \(0\neq s\in S\), there exist a positive integer \(n\) and a left ideal \(X_{s^n}\) of \(S\) such that \(s^n\neq 0\) and \(\mathbf l_S(\text{Ker}(s^n))=Ss^n\oplus X_{s^n}\). Some characterizations and properties of AGQP-injective modules are given, and some properties of AGQP-injective modules with additional conditions are studied.

### MSC:

16D50 | Injective modules, self-injective associative rings |

### Keywords:

almost general quasi-principally injective modules; AGQP-injective modules; generalizations of injectivity### Software:

AGQP
PDF
BibTeX
XML
Cite

\textit{Z. Zhu} and \textit{X. Zhang}, Int. J. Math. Math. Sci. 2008, Article ID 469725, 7 p. (2008; Zbl 1160.16002)

### References:

[1] | W. K. Nicholson and M. F. Yousif, “Principally injective rings,” Journal of Algebra, vol. 174, no. 1, pp. 77-93, 1995. · Zbl 0839.16004 |

[2] | S. B. Nam, N. K. Kim, and J. Y. Kim, “On simple GP-injective modules,” Communications in Algebra, vol. 23, no. 14, pp. 5437-5444, 1995. · Zbl 0840.16006 |

[3] | N. V. Sanh, K. P. Shum, S. Dhompongsa, and S. Wongwai, “On quasi-principally injective modules,” Algebra Colloquium, vol. 6, no. 3, pp. 269-276, 1999. · Zbl 0949.16003 |

[4] | R. Yue Chi Ming, “On injectivity and p-injectivity,” Journal of Mathematics of Kyoto University, vol. 27, no. 3, pp. 439-452, 1987. · Zbl 0655.16012 |

[5] | J. Chen, Y. Zhou, and Z. Zhu, “GP-injective rings need not be P-injective,” Communications in Algebra, vol. 33, no. 7, pp. 2395-2402, 2005. · Zbl 1076.16003 |

[6] | S. S. Page and Y. Zhou, “Generalizations of principally injective rings,” Journal of Algebra, vol. 206, no. 2, pp. 706-721, 1998. · Zbl 0923.16002 |

[7] | Z. Zhu, “On general quasi-principally injective modules,” Southeast Asian Bulletin of Mathematics, vol. 30, no. 2, pp. 391-397, 2006. · Zbl 1113.16008 |

[8] | W. K. Nicholson, J. K. Park, and M. F. Yousif, “Principally quasi-injective modules,” Communications in Algebra, vol. 27, no. 4, pp. 1683-1693, 1999. · Zbl 0924.16004 |

[9] | N. V. Dung, D. V. Huynh, P. F. Smith, and R. Wisbauer, Extending Modules, vol. 313 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1994. · Zbl 0841.16001 |

[10] | M. F. Yousif and Y. Zhou, “Rings for which certain elements have the principal extension property,” Algebra Colloquium, vol. 10, no. 4, pp. 501-512, 2003. · Zbl 1040.16004 |

[11] | S. K. Jain and S. R. López-Permouth, “Rings whose cyclics are essentially embeddable in projective modules,” Journal of Algebra, vol. 128, no. 1, pp. 257-269, 1990. · Zbl 0698.16012 |

[12] | Y. Zhou, “Rings in which certain right ideals are direct summands of annihilators,” Journal of the Australian Mathematical Society, vol. 73, no. 3, pp. 335-346, 2002. · Zbl 1020.16003 |

[13] | J. Chen and N. Ding, “On regularity of rings,” Algebra Colloquium, vol. 8, no. 3, pp. 267-274, 2001. · Zbl 0991.16004 |

[14] | R. Wisbauer, Foundations of Module and Ring Theory, vol. 3 of Algebra, Logic and Applications, Gordon and Breach Science, Philadelphia, Pa, USA, German edition, 1991. · Zbl 0746.16001 |

[15] | R. Wisbauer, M. F. Yousif, and Y. Zhou, “Ikeda-Nakayama modules,” Contributions to Algebra and Geometry, vol. 43, no. 1, pp. 111-119, 2002. · Zbl 1007.16003 |

[16] | Z. Zhu, Z. Xia, and Z. Tan, “Generalizations of principally quasi-injective modules and quasiprincipally injective modules,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 12, pp. 1853-1860, 2005. · Zbl 1089.16006 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.