Let M be a finitely generated quasi-projective module. For any such Mwe prove the following results:

(i) Every quotient module N of M is is strongly p-regular.

(ii) Every quotient module N of M is simultaneously Hopfian and is strongly p-regular for every quotient N of M.

We give an example of a non-finitely generated quasi-projective abelian group A with every factor group of A simultaneously Hopfian and co-Hopfian but not strongly p-regular. We also introduce the concept of a primitive module and show that if L is any module satisfying with all primitive quotients of L Hopfian, then L itself is Hopfian. In particular any V-module L with Hopfian primitive quotients is itself Hopfian.

Hopfian modules, co-Hopfian modules, strongly p-regular rings,V-modules.