Published online by Cambridge University Press: 24 June 2010
Motivated by a relation on submodules of a module used by both A. W. Goldie and P. F. Smith, we say submodules X, Y of M are β* equivalent, Xβ*Y, if and only if is small in and is small in . We show that the β* relation is an equivalence relation and has good behaviour with respect to addition of submodules, homomorphisms and supplements. We apply these results to introduce the class of
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