ON CO-NOETHERIAN DIMENSION OF RINGS

We define and study co-Noetherian dimension of rings for which the injective envelope of simple modules have finite Krulldimension. 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 Wn of rings with co-Noetherian dimension ≤ 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.