WHEN DO THE DIRECT SUMS OF MODULES INHERIT CERTAIN PROPERTIES?

It is of obvious interest to know whether an algebraic property of modules is preserved by direct sums of such modules. In this paper we provide a survey of this question for various classes of modules of interest. The question of inheritance of a property by direct sums of modules has been explored for the classes of (quasi-)injective modules and some of their generalizations as a motivation for further work. In the main part of this paper we provide latest results and developments on this question for the related classes of Baer, quasi-Baer, and Rickart modules. Examples are provided that delimit our results and explain the notions. Some open problems are listed at the end of the paper.