On co-Noetherian dimension of rings

Document Type : Research Paper


Isfahan University of Technology


We define and study
co-Noetherian dimension of rings for which the injective envelope
of simple modules have finite Krull-dimension. This  is a Morita
invariant dimension that measures how far the ring is from being
co-Noetherian. The co-Noetherian dimension of certain rings,
including commutative rings, are determined. It is
 shown that the class ${\mathcal W}_n$ of rings with co-Noetherian dimension $\leq
n$ is closed under homomorphic images and finite normalizing
extensions, and that for each $n$ there exist rings with
co-Noetherian dimension $n$. The possible relations between Krull
 and co-Noetherian dimensions  are investigated, and examples are provided to
 show that these
 dimensions are independent of each