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 $R_R$ 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].