Noetherian rings are those having ascending chain condition on ideals. There is also literature concerning ACC on n-generated (i.e., generated by n elements) ideals; see e.g, Commutative Rings with ACC on n-Generated Ideals, JOURNAL OF ALGEBRA 80, 261-278 (1983). What will happen if a ring R satisfies that each R-modules has ACC on chains of finitely generated submodules (that is, every ascending chain of submodules of any module M, each of which is finitely generated, must stabilize)?
Ring satisfying the ascending chain condition on right ideals is clearly right Noetherian.
If a ring contains a right ideal that isn't finitely generated, then you can construct a strictly ascending chain of finitely generated right ideals within it.
Note that nonzero rings will always have modules not satisfying ACC on on f.g. submodules, so it is not very interesting to require it on all modules.