Generalized triangular matrix rings and the fully invariant extending property

A module M is called (strongly) FI-extending if every fully invariant submodule of M is essential in a (fully invariant) direct summand of M. A ring R with unity is called quasi-Baer if the right annihilator of every ideal is generated, as a right ideal, by an idempotent. For semi-prime rings the FI-extending condition, strongly FI-extending condition and quasi-Baer condition are equivalent. In this paper we fully characterize the 2-by-2 generalized (or formal) triangular matrix rings which are either (right) FI-extending, (right) strongly FI-extending, or quasi-Baer. Examples are provided to illustrate and delimit our results.