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**Rings whose cyclics are essentially embeddable in projective modules.**
*(English)*
Zbl 0698.16012

The authors call a ring \(R\) a right (left) CEP-ring if each cyclic right (left) \(R\)-module is essentially embeddable in a projective module. Quasi-Frobenius (QF) rings and right uniserial rings are examples of right CEP-rings and in fact a theorem of C. Faith and E. A. Walker [J. Algebra 5, 203-221 (1967; Zbl 0173.03203)] shows that \(R\) is QF if and only if \(R\) is both a right and a left CEP-ring. Here it is shown that if \(R\) is both a QF-3 ring and a right CEP-ring then \(R\) is QF. (Recall that \(R\) is said to be QF-3 if its right injective hull is projective.) Moreover it is shown that a semiperfect ring \(R\) is right CEP if and only if (i) \(R\) is right Artinian and (ii) every indecomposable projective module is both uniform and weakly \(R\)-injective. (Here a module \(P\) is called weakly \(R\)-injective if, given any \(f\in\operatorname{Hom}(R,E(P))\), where \(E(P)\) is the injective hull of \(P\), then \(f(1)\) belongs to some submodule \(X\) of \(E(P)\) isomorphic to \(P\). Also, as a “Note added in proof”, the authors say that the uniform assumption may be omitted in this characterization.) Another main result characterizes those rings \(R\) of which every homomorphic image is a right CEP-ring as being a direct sum of right uniserial rings or matrix rings over right self-injective right uniserial rings. The paper ends with illustrative examples obtained using the trivial (split) extension construction.

Reviewer: J.Clark

### MSC:

16L60 | Quasi-Frobenius rings |

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

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

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

### Keywords:

trivial extension; quasi-Frobenius rings; right uniserial rings; right CEP-rings; left CEP-ring; QF-3 ring; right injective hull; semiperfect ring; indecomposable projective module; direct sum of right uniserial rings; matrix rings over right self-injective right uniserial rings### Citations:

Zbl 0173.03203
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\textit{S. K. Jain} and \textit{S. R. López-Permouth}, J. Algebra 128, No. 1, 257--269 (1990; Zbl 0698.16012)

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### References:

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

[2] | Faith, C., Algebra: Rings, Modules and Categories I (1973), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg · Zbl 0266.16001 |

[3] | Faith, C., Algebra: Rings, Modules and Categories II (1976), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg |

[4] | Faith, C., Self-injective rings, (Proc. Amer. Math. Soc., 77 (1979)), 157-164 · Zbl 0424.16014 |

[5] | Faith, C., Embedding modules in projectives: A report on a problem, (Advances in Non-commutative Ring Theory. Advances in Non-commutative Ring Theory, Lectures Notes in Mathematics, Vol. 951 (1981), Springer-Verlag: Springer-Verlag New York/Berlin/Heidelberg) · Zbl 0504.16009 |

[6] | Faith, C.; Walker, E. A., Direct sum representation of injective modules, J. Algebras, 5, 203-221 (1967) · Zbl 0173.03203 |

[7] | Jain, S. K.; López-Permouth, S. R., A generalization of the Wedderburn-Artin theorem, (Proc. Amer. Math. Soc., 106 (1987)), 19-23 · Zbl 0681.16016 |

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