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On covers. (English) Zbl 0973.16002
Let \(R\) 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.

MSC:
16D40 Free, projective, and flat modules and ideals in associative algebras
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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