How Far is a Ring from Being Co-Noetherian? Sept. 26

The Ohio State University-Ohio University Ring Theory Seminar meets on Friday, Sept. 26, at 4:45 p.m. in Cockins Hall 240 at OSU.

The speaker is Dr. Ahmad Haghny, Professor of Mathematics at the Isfahan University of Technology in Iran. His talk is on “How Far is a Ring from Being Co-Noetherian?”

Abstract: Let W(0) denote the class of co-Noetherian rings, and for each positive integer n, denote by W(n), the class of rings whose finitely cogenerated modules have Krull- dimension (in the sense of Rentsler and Gabriel) not exceeding n. We study these classes as well as the class W of rings R for which each E(S) has Krull- dimension where S is a simple R-module.

The classes W(n) form a proper chain, and all the above classes are closed under factor rings, Morita equivalence and finite normalizing extensions.

If a ring R is in W(n) but not in W(n-1), we say that R has co-Noetherian dimension n. The co-Notherian dimension does not exceed the Krull dimension, when they both exist. We determine the co-Noetherian dimension of commutative rings and prove that a semi Artinan ring R is co-Noetherian when R is not in W. Various examples are provided to show the independence of co-Noetherian dimension and Krull dimension from each other. We end the talk with a discussion of generalization of the above results and we will point out to further research in this area. (This is a joint work with M. R. Vedadi).