When a (dual-)Baer module is a direct sum of (co-)prime modules

Since 2004, Baer modules have been considered by many authors as a generalization of the Baer rings. A module $M_R$ is called Baer if every intersection of the kernels of endomorphisms on $M_R$ is a direct summand of $M_R$. It is known that commutative Baer rings are reduced. We prove that if a Baer module $M$ is a direct sum of prime modules, then every direct summand of $M$ is retractable. The converse is true whenever the triangulating dimension of $M$ is finite (e.g. if the uniform dimension of $M$ is finite). Dually, if every direct summand of a dual-Baer module $M$ is co-retractable, then it is a direct sum of co-prime modules and the converse is true whenever the sum is finite or $M$ is a max-module. Among other applications, we show that if $R$ is a commutative hereditary Noetherian ring then a finitely generated $R$-module is Baer iff it is projective or semisimple. Also, over a ring Morita equivalent to a perfect duo ring, all dual-Baer modules are semisimple.