We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subsetA such that 2|A| = 2|B|. This implies in particular that B has 2|B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from...

In this paper we introduce the notion of e-gH modules which is a proper generalization Hopfian and defined as, module called if, any surjective -endomorphism has an e-small kernel, ring e-gH. We give some characterizations properties modules.