Commutative Tall Rings

Abstract

A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.

On commutative rings whose maximal ideals are idempotent

“…Recall that a module M is called tall if it contains a submodule N such that both M/N and N are non-noetherian. A ring R is called tall if every non-noetherian R-module is tall (for example, max rings are tall by[18, Corollary 1.2]). In 1976, B. Sarath showed that the class of rings R for which every module having Krull dimension is noetherian is exactly that of tall rings (see[19, Theorem 2.7]).…”