On covers.

*(English)*Zbl 0973.16002Let \(R\) be a ring. For a class \(\mathcal C\) of \(R\)-modules closed under isomorphisms, a homomorphism \(\psi\colon C\to M\) with \(C\in{\mathcal C}\) is called a \(\mathcal C\)-precover of \(M\) if, for each homomorphism \(f\colon F\to M\) with \(F\in{\mathcal C}\), there is a homomorphism \(g\colon F\to C\) such that \(\psi g=f\); \(\psi\) is said to be a \(\mathcal C\)-cover if, in addition, every endomorphism \(h\) of \(G\) with \(\psi h=\psi\) is an automorphism. The main results give some cardinal/module theoretic conditions for \(\mathcal C\)-covers to exist. This general theory is applied to give some partial existence results for flat covers (i.e., \(\mathcal C=\) flat modules), a problem which has been completely solved in a subsequent paper by E. E. Enochs, R. El Bashir, and L. Bican [All modules have flat covers (to appear)]. The general results are also applied to give some information on the existence of \(\tau\)-torsionfree covers (i.e., \({\mathcal C}=\tau\)-torsionfree modules) when \(\tau\) is a hereditary torsion theory for \(R\)-mod.

Reviewer: Mark L.Teply (Milwaukee)

##### MSC:

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

16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |

##### Keywords:

precovers; covers; torsionfree covers; homomorphisms; endomorphisms; automorphisms; flat covers; flat modules; hereditary torsion theories
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\textit{L. Bican} and \textit{B. Torrecillas}, J. Algebra 236, No. 2, 645--650 (2001; Zbl 0973.16002)

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