# Rings where finitely generated ideals are closed under countable intersection

Does there exist a characterization of those rings $$R$$ such that finitely generated left ideals are closed under countable intersection?

For example, any noetherian ring has this property, since all ideals are finitely generated. On the other hand, being coherent is not quite sufficient, as this counterexample shows.

More generally, I'd be interested in sufficient conditions on coherent rings which imply that they have this property.

• I'm not optimistic that we can say much here. Commented Nov 11, 2023 at 17:18

Namely, assume that $$R$$ is graded (by non-negative integers) algebra (with 1) over a field which is finitely generated in positive degrees (like the tensor algebra $$TV$$ of a finite-dimensional vector space $$V$$). Then $$R$$ is generated by a finite-dimensional homogeneous vector space $$V\subset R_{\ge 1}$$. If $$R$$ is non-noetherian, there is an infinite-generated left ideal $$I$$. We may assume the it is (minimally) generated by a sequence $$a_1, a_2, \dots$$ of elements with non-decreasing degrees $$d_i = \deg a_i$$. Then $$I$$ is the intersection of the ideals $$I_i = (a_1, \dots, a_{i}) + (V^{d_{i+1}})$$ for $$i=1,2,\dots$$, that are obviously finitely generated.