zbMATH — the first resource for mathematics

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)
Full Text:
References:
 [1] Bass, H., Finitistic dimension and a homological characterization of semi-primary rings, Trans. amer. math. soc., 95, 466-488, (1960) · Zbl 0094.02201 [2] L. Bican, A note on pure subgroups, to appear. · Zbl 0969.20028 [3] L. Bican, Pure subgroups, to appear. · Zbl 0983.20054 [4] Eckmann, B.; Schopf, A., Über injektive moduln, Archiv. math., 4, 75-78, (1953) · Zbl 0050.25904 [5] Enochs, E., Injective and flat covers, envelopes and resolvents, Israel J. math., 39, 189-209, (1981) · Zbl 0464.16019 [6] Fuchs, L., Infinite abelian groups, (1973 and 1977), Academic Press New York [7] Teply, M.L., Torsion-free covers, II, Israel J. math., 23, 132-136, (1976) · Zbl 0321.16014 [8] Xu, J., Flat covers of modules, Lecture notes in mathematics, 1634, (1996), Springer-Verlag Berlin/Heidelberg/New York
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.