On modules for which the finite exchange property implies the countable exchange property

It has been a long standing open problem whether the finite exchange property implies the full exchange property for an arbitrary module. The main results of this paper are Theorem 1.1: For modules whose idempotent endomorphisms are central, the finite exchange property implies the countable exchange property, and Theorem 2.11: Over a ring with ace on essential right ideals, the finite exchange property implies the full exchange property for every quasi-continuous module. The latter can be viewed as a partial affirmative answer to an open problem of Mohamed and Muller [8].