It is proved that if R is a right and left Noetherian ring then the right R-module RI satisfies the ascending chain condition on n-generated submodules, for every positive integer n. Mathematics Subject Classification (2000): 16P70

We introduce the notion of Burch submodules and weakly m-full modules over a local ring (R,m) study their properties. One our main results shows that satisfy 2-Tor rigid test also show (R,m), submodule M finitely generated R-module X, such either M=mX or M(⊆mX) is in 1-Tor rigid, module provided X faithful (and X∕M has finite length when m-full). As an application, we give some new class rings ...