On co-Noetherian dimension of rings

We define and study co-Noetherian dimension of rings for which the injective envelopeof 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 other.