## Abstract

This paper studies the existence and properties of a torsion-free cover with respect to a faithful hereditary torsion theory (T, F) of modules over a ring with unity. A direct sum of a finite number of torsion-free covers of modules is the torsion-free cover of the direct sum of the modules. The concept of a T-near homomorphism, which generalizes Enochs' definition of a neat submodule, is introduced and studied. This allows the generalization of a result of Enochs on liftings of homomorphisms. Hereditary torsion theories for which every module has a torsion-free cover are called universally covering. If the inclusion map of R into the appropriate quotient ring Q is a left localization in the sense of Silver, the problem of the existence of universally-covering torsion theories can be reduced to the case R=Q. As a consequence, many sufficient conditions for a hereditary torsion theory to be universally covering are obtained. For a universally-covering hereditary torsion theory (T, F), the following conditions are equivalent: (1) the product of F-neat homomorphisms is always T-neat; (2) the product of torsion-free covers is always T-neat; (3) every nonzero module in T has a nonzero socle.

Original language | English |
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Pages (from-to) | 237-256 |

Number of pages | 20 |

Journal | Israel Journal of Mathematics |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1973 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics