##
**Generalizations of principally injective rings.**
*(English)*
Zbl 0923.16002

Given a module \(M_R\), denote \(S=\text{End}_R(M)\). The module \(M\) is said to be almost principally injective (or simply AP-injective) if, for any \(a\in R\), there is an \(S\)-submodule \(X_a\) of \(M\) such that \(\ell_M(r_R(a))=a\oplus X_a\) as left \(S\)-modules. The module \(M\) is called almost general principally injective (or simply AGP-injective) if, for any \(0\neq a\in R\), there is a positive integer \(n(a)\), depending on \(a\), and an \(S\)-submodule \(X_a\) of \(M\) such that \(a^n\neq 0\) and \(\ell_M(r_R(a))=Ma^n\oplus X_a\) as left \(S\)-modules. If \(R_R\) is an AP-injective (resp. AGP-injective) module, we call \(R\) a right AP-injective (resp. right AGP-injective) ring. Furthermore, a ring \(R\) is called a right QP-injective ring if, for any \(0\neq a\in R\), there is a left ideal \(X_a\) such that \(\ell_R(r_R(a))=Ra+X_a\) with \(a\not\in X_a\). More generally, a ring \(R\) is called right QGP-injective if, for each \(0\neq a\in R\), there is a positive integer \(n=n(a)\) depending on \(a\) such that \(\ell_R(r_R(a))=Ra^n+X_a\) such that \(a^n\not\in X_a\).

After providing several interesting examples which show the AP-injectivity and the GP-injectivity are distinct, the relationship between the Jacobson radical \(J(R)\) and the right singular ideal \(Z(R_R)\) is investigated. For example, it is shown that if \(R\) is a right AGP-injective ring, then \(J(R)=Z(R_R)\). Thereby results of S. B. Nam, N. K. Kim and J. Y. Kim [Commun. Algebra 23, No. 14, 5437-5444 (1995; Zbl 0840.16006)] and W. K. Nicholson and M. F. Yousif [J. Algebra 174, No. 1, 77-93 (1995; Zbl 0839.16004)] can be extended. Also it is proved that if \(R\) is semiperfect right QGP-injective, then \(\text{Soc}(R_R)\subseteq\text{Soc}({_RR})\) and \(Z({_RR})\subseteq Z(R_R)=J(R)\).

For the nilpotency of \(J(R)\), it is proved that \(J(R)\) is nilpotent if \(R\) is right AGP-injective and \(R/\text{Soc}(R_R)\) satisfies the ACC on right annihilators. This result extends a result of E. P. Armendariz and J. K. Park [Arch. Math. 58, No. 1, 24-33 (1992; Zbl 0768.16001)].

A decomposition of a right QGP-injective ring with an additional assumption into a semisimple ring and a ring with square zero right socle is investigated. Also necessary and sufficient conditions for a matrix ring to be an AP-injective ring are considered.

After providing several interesting examples which show the AP-injectivity and the GP-injectivity are distinct, the relationship between the Jacobson radical \(J(R)\) and the right singular ideal \(Z(R_R)\) is investigated. For example, it is shown that if \(R\) is a right AGP-injective ring, then \(J(R)=Z(R_R)\). Thereby results of S. B. Nam, N. K. Kim and J. Y. Kim [Commun. Algebra 23, No. 14, 5437-5444 (1995; Zbl 0840.16006)] and W. K. Nicholson and M. F. Yousif [J. Algebra 174, No. 1, 77-93 (1995; Zbl 0839.16004)] can be extended. Also it is proved that if \(R\) is semiperfect right QGP-injective, then \(\text{Soc}(R_R)\subseteq\text{Soc}({_RR})\) and \(Z({_RR})\subseteq Z(R_R)=J(R)\).

For the nilpotency of \(J(R)\), it is proved that \(J(R)\) is nilpotent if \(R\) is right AGP-injective and \(R/\text{Soc}(R_R)\) satisfies the ACC on right annihilators. This result extends a result of E. P. Armendariz and J. K. Park [Arch. Math. 58, No. 1, 24-33 (1992; Zbl 0768.16001)].

A decomposition of a right QGP-injective ring with an additional assumption into a semisimple ring and a ring with square zero right socle is investigated. Also necessary and sufficient conditions for a matrix ring to be an AP-injective ring are considered.

Reviewer: J.K.Park (Pusan)

### MSC:

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

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

### Keywords:

AP-injectivity; GP-injectivity; Jacobson radical; right singular ideals; right AGP-injective rings; ACC on right annihilators; right QGP-injective rings; right socles; AP-injective rings
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\textit{S. S. Page} and \textit{Y. Zhou}, J. Algebra 206, No. 2, 706--721 (1998; Zbl 0923.16002)

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### References:

[1] | Anderson, F. W.; Fuller, K. R., Rings and Categories of Modules (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0242.16025 |

[2] | Armendariz, E. P.; Park, J. K., Self-injective rings with restricted chain conditions, Arch. Math., 58, 24-33 (1992) · Zbl 0768.16001 |

[3] | Camillo, V., Commutative rings whose principal ideals are annihilators, Protugal. Math., 46, 33-37 (1989) · Zbl 0668.13007 |

[4] | Fisher, J. W., On the nilpotency of nil subrings, Canad. J. Math., 22, 1211-1216 (1970) · Zbl 0194.34602 |

[5] | Jain, S. K.; López-Permouth, S. R., Rings whose cyclics are essentially embeddable in projectives, J. Algebra, 128, 257-269 (1990) · Zbl 0698.16012 |

[6] | Koh, K., On some characteristic properties of self-injective rings, Proc. Amer. Math. Soc., 19, 209-213 (1968) · Zbl 0155.07205 |

[7] | Nam, S. B.; Kim, N. K.; Kim, J. Y., On simple GP-injective modules, Comm. Algebra, 14, 5437-5444 (1995) · Zbl 0840.16006 |

[8] | Mohamed, S. H.; Müller, B. J., Continuous and Discrete Modules (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0701.16001 |

[9] | Nicholson, W. K.; Yousif, M. F., Principally injective rings, J. Algebra, 174, 77-93 (1995) · Zbl 0839.16004 |

[10] | Nicholson, W. K.; Yousif, M. F., On a theorem of Camillo, Comm. Algebra, 14, 5309-5314 (1995) · Zbl 0839.16005 |

[11] | Yousif, M. F., On semiperfect FPF-rings, Canad. Math. Bull., 37, 287-288 (1994) · Zbl 0816.16004 |

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