Rings for which f.g. projective modules have the FI-extending property

A right R-module M over a ring R is said to be FI-extending if any fully invariant submodule of M is essential in a direct summand of M. We prove that if R has ACC on the right annihilators, then RR is FI-extending if, and only if, every f.g. projective module over R is FI-extending. This is an affirmative answer to the question raised by Birkenmeier–Park–Rizvi [Comm. Algebra 30 (2002), 1833–1852].