Covers, precovers, and purity

TY - JOUR

T1 - Covers, precovers, and purity

AU - Holm, Henrik

AU - Peter, Jørgensen

PY - 2008

Y1 - 2008

N2 - We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it forms the first half of a so-called perfect cotorsion pair as introduced by Salce; this is stronger than being covering. Some applications are given to concrete classes of modules such as kernels of homological functors and torsion free modules in a torsion pair.

AB - We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it forms the first half of a so-called perfect cotorsion pair as introduced by Salce; this is stronger than being covering. Some applications are given to concrete classes of modules such as kernels of homological functors and torsion free modules in a torsion pair.

UR - http://www.scopus.com/inward/record.url?scp=77951465056&partnerID=8YFLogxK

U2 - 10.1215/ijm/1248355359

DO - 10.1215/ijm/1248355359

M3 - Journal article

AN - SCOPUS:77951465056

SN - 0019-2082

VL - 52

SP - 691

EP - 703

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

IS - 2

ER -