turkmath.org

A ring is said to be right tall if every non-noetherian right module contains a proper non-noetherian submodule. It is easy to see that every right max ring is right tall, however there are examples of commutative non-tall rings which are max. We give several ring-theoretical criteria of commutative tall rings in general and in some particular cases. It will be proved that a commutative ring is tall if and only if there exists no noetherian artinian module over it, and that commutative noetherian tall rings are precisely artinian ones.