π-dual Baer Modules and π-dual Baer Rings

Let R be a ring and let M be an R-module with S = End_R (M). A submodule N of M is said to be projection invariant in M (denoted N ⊴_p M) if eN ⊆ N for all e = e² ∈ S. We call M π-dual Baer, if for each N ⊴_p M there exists e² = e ∈ S such that {f ∈ S | f (M) ⊆ N} = eS. A characterization of π-dual Baer modules is provided. We show that the class of π-dual Baer modules lies strictly between the classes of dual Baer modules and quasi-dual Baer modules. It is also shown that in general, the class of π-dual Baer modules is neither closed under direct sums nor closed under direct summands. The structure of π-dual Baer modules over Dedekind domains is completely determined. We conclude the paper by studying right π-dual Baer rings. We call a ring R right π-dual Baer if the right R-module R_R is right π-dual Baer. A characterization of this class of rings is provided. We also investigate the transfer between a base ring R and many of its extensions (for example, full matrix rings over R or R[x] or R[[x]]). In addition, we characterize the 2-by-2 generalized triangular right π-dual Baer matrix rings.