GOLDIE-SUPPLEMENTED MODULES

摘要

Motivated by a relation on submodules of a module used by bothnA.W. Goldie and P. F. Smith, we say submodules X,Y ofM are β∗ equivalent, Xβ∗Y,nif and only ifnX+YnX is small in MnX and X+YnY is small in MnY .We showthat the β∗ relation is annequivalence relation and has good behaviour with respect to addition of submodules,nhomomorphisms and supplements. We apply these results to introduce the class ofnG∗-supplementedmodules and to investigate this class and the class ofH-supplementednmodules. These classes are located among various well-known classes of modulesnrelated to the class of lifting modules. Moreover these classes are used to explore annopen question of S.H.Mohamed and B. J.Mueller. Examples are provided to illustratenand delimit the theory.