On direct sums of Baer modules

The notion of Baer modules was defined recently. Since a direct sum of Baer modules is not a Baer module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be Baer. In this paper we study rings for which every free module is Baer. It is shown that this is precisely the class of semiprimary hereditary rings. We also prove that every finite rank free R-module is Baer if and only if R is right semihereditary, left Π-coherent. Necessary and sufficient conditions for finite direct sums of copies of a Baer module to be Baer are obtained, for the case when M is retractable. An example of a module M is exhibited for which Mn is Baer but Mn+1 is not Baer. Other results on direct sums of Baer modules to be Baer under some additional conditions are obtained. Some applications are also included.