  In this paper, we study module theoretic definitions of the Baer and related
ring concepts. We say a module is s.Baer if the right annihilator of a nonempty
subset of the module is generated by an idempotent in the ring. We show that
s.Baer modules satisfy a number of closure properties. Under certain
conditions, a torsion theory is established for the s.Baer modules, and we
provide examples of s.Baer torsion modules and modules with a nonzero s.Baer
radical. The other principal interest of this paper is to provide explicit
connections between s.Baer modules and projective modules. Among other results,
we show that every s.Baer module is an essential extension of a projective
module. Additionally, we prove, with limited and natural assumptions, that in a
generalized triangular matrix ring every s.Baer submodule of the ring is
projective. As an application, we show that every prime ring with a minimal
right ideal has the strong summand intersection property. Numerous examples are
provided to illustrate, motivate, and delimit the theory.
