##
**Mininjective rings.**
*(English)*
Zbl 0879.16002

For a ring \(R\), a right \(R\)-module \(M\) is called mininjective if for each simple right ideal \(I\) of \(R\) every \(R\)-homomorphism \(f\colon I\to M\) extends to \(R\). A ring \(R\) is called right mininjective if \(R_R\) is mininjective; equivalently every isomorphism between simple right ideals is given by left multiplication by an element of \(R\). Mininjective rings were initially introduced by M. Harada [Osaka J. Math. 19, 587-597 (1982; Zbl 0495.16022)] and were studied for the Artinian case by M. Harada [loc.cit., and ibid. 20, 1-4 (1983; Zbl 0516.16009)]. A module is said to have squarefree socle if every nonzero homogeneous component of its socle is simple. A ring \(R\) is called right min-PF if \(R\) is semiperfect, right mininjective in which \(\text{Soc}(R_R)\) is essential in \(R_R\) and \(\ell(r(K))=K\) of every simple left ideal \(K\subseteq Re\), where \(e\) is a local idempotent.

It is shown that the right mininjectivity is a Morita invariant property, and a commutative ring is mininjective if and only if it has squarefree socle. Semiperfect right mininjective rings are completely characterized: a semiperfect ring \(R\) is right mininjective if and only if \(\text{Soc}(R_R)e\) is simple or zero for each local idempotent \(e\) of \(R\).

Furthermore several interesting results for right mininjectivity, which can extend well known results on pseudo- and quasi-Frobenius rings, are investigated. For examples: (1) If \(R\) is a semiperfect, right mininjective ring for which \(\text{Soc}(eR)\neq 0\) for every local idempotent \(e\) of \(R\), then \(R\) is right and left Kasch, and \(R\) admits a Nakayama permutation of its basic idempotents; (2) If \(R\) is a right min-PF, then \(R\) is right and left Kasch, \(J(R)=Z(R_R)\), and \(R\) admits a Nakayama permutation of its basic idempotents, where \(J(R)\) and \(Z(R_R)\) denote the Jacobson radical and the right singular ideal of \(R\), respectively. Also as an application, it is proved that a right Artinian ring \(R\) is quasi-Frobenius if and only if \(R\) is mininjective.

It is shown that the right mininjectivity is a Morita invariant property, and a commutative ring is mininjective if and only if it has squarefree socle. Semiperfect right mininjective rings are completely characterized: a semiperfect ring \(R\) is right mininjective if and only if \(\text{Soc}(R_R)e\) is simple or zero for each local idempotent \(e\) of \(R\).

Furthermore several interesting results for right mininjectivity, which can extend well known results on pseudo- and quasi-Frobenius rings, are investigated. For examples: (1) If \(R\) is a semiperfect, right mininjective ring for which \(\text{Soc}(eR)\neq 0\) for every local idempotent \(e\) of \(R\), then \(R\) is right and left Kasch, and \(R\) admits a Nakayama permutation of its basic idempotents; (2) If \(R\) is a right min-PF, then \(R\) is right and left Kasch, \(J(R)=Z(R_R)\), and \(R\) admits a Nakayama permutation of its basic idempotents, where \(J(R)\) and \(Z(R_R)\) denote the Jacobson radical and the right singular ideal of \(R\), respectively. Also as an application, it is proved that a right Artinian ring \(R\) is quasi-Frobenius if and only if \(R\) is mininjective.

Reviewer: J.K.Park (Pusan)

### MSC:

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

16L60 | Quasi-Frobenius rings |

16L30 | Noncommutative local and semilocal rings, perfect rings |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16P20 | Artinian rings and modules (associative rings and algebras) |

### Keywords:

semiperfect rings; simple right ideals; local idempotents; right mininjectivity; squarefree socles; quasi-Frobenius rings; Nakayama permutations; basic idempotents; left Kasch rings; right Artinian rings
PDF
BibTeX
XML
Cite

\textit{W. K. Nicholson} and \textit{M. F. Yousif}, J. Algebra 187, No. 2, 548--578 (1997; Zbl 0879.16002)

Full Text:
DOI

### References:

[1] | Ara, P.; Park, J. K., On continuous semiprimary rings, Comm. Algebra, 19, 1945-1957 (1991) · Zbl 0732.16018 |

[2] | Baba, Y.; Oshiro, K., On a theorem of Fuller, J. Algebra, 154, 86-94 (1993) · Zbl 0808.16022 |

[3] | Beachy, J. A., On quasi-artinian rings, J. London Math. Soc., 3, 449-452 (1971) · Zbl 0214.05603 |

[4] | Björk, J.-E., Rings satisfying certain chain conditions, J. Reine Angew. Math., 245, 63-73 (1970) · Zbl 0211.36401 |

[5] | Camillo, V., Distributive modules, J. Algebra, 36, 16-25 (1975) · Zbl 0308.16015 |

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

[7] | Camillo, V.; Yousif, M. F., Continuous rings with acc on annihilators, Canad. Math. Bull., 34, 462-464 (1991) · Zbl 0767.16004 |

[8] | Curtis, C. W.; Reiner, I., Representation Theory of Finite Groups and Associative Algebras (1962), Wiley-Interscience: Wiley-Interscience New York/London · Zbl 0131.25601 |

[9] | Dieudonné, J., Remarks on quasi-Frobenius rings, Illinois J. Math., 2, 345-354 (1958) · Zbl 0101.02701 |

[10] | Faith, C., Algebra II, Ring Theory (1976), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0335.16002 |

[11] | Fuller, K. R., On indecomposable injectives over artinian rings, Pacific J. Math., 29, 115-135 (1969) · Zbl 0182.05702 |

[12] | Gordon, R., Rings in which minimal left ideals are projective, Pacific J. Math., 31, 679-692 (1969) · Zbl 0188.08402 |

[13] | Hajarnavis, C. R.; Norton, N. C., On dual rings and their modules, J. Algebra, 93, 253-266 (1985) · Zbl 0595.16009 |

[14] | Harada, M., On modules with extending properties, Osaka J. Math., 19, 203-215 (1982) · Zbl 0491.16026 |

[15] | Harada, M., Self mini-injective rings, Osaka J. Math., 19, 587-597 (1982) · Zbl 0495.16022 |

[16] | Harada, M., A characterization of QF-algebras, Osaka J. Math., 20, 1-4 (1983) · Zbl 0516.16009 |

[17] | Kasch, F., London Mathematical Society Monographs, Modules and Rings (1982), Academic Press: Academic Press New York |

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

[19] | Nicholson, W. K.; Yousif, M. F., Continuous rings with chain conditions, J. Pure Appl. Algebra, 97, 325-332 (1994) · Zbl 0824.16020 |

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

[21] | Osofsky, B. L., A generalization of quasi-Frobenius rings, J. Algebra, 4, 373-389 (1966) · Zbl 0171.29303 |

[22] | Rutter, E. A., Rings with the principal extension property, Comm. Algebra, 3, 203-212 (1975) · Zbl 0298.16015 |

[23] | Storrer, H. H., A note on quasi-Frobenius rings and ring epimorphisms, Canad. Math. Bull., 12, 287-291 (1969) · Zbl 0182.36704 |

[24] | Utumi, Y., On continuous rings and self-injective rings, Trans. Amer. Math. Soc., 118, 158-173 (1965) · Zbl 0144.27301 |

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.