Ascending chain conditions in groups, rings and modules

The chain conditions in question are of the following type: given a positive integer n, an algebraic object satisfies the ascending chain condition on n-generated subobjects. For example, every free group satisfies the ascending chain condition on n-generated subgroups, for every positive integer n. If R is a right Noetherian ring then sometimes every free right R-module satisfies the ascending chain condition on n-generated submodules, for every positive integer n. This is a very non-technical talk intended for a wide audience. All you need to know is what "Noetherian" means and some simple facts about Abelian groups.