Extension closure of k-torsionfree modules

Let Lambda be a left and right Noetherian ring. For a positive integer k, w e give an equivalent condition that flat dimensions of the first k terms in the minimal injective resolution of Lambda are less than or equal to k. In this case we show that the subcategory consisting of k-torsionfree modules is extension closed. Moreover we prove that for a Noetherian algebra every subcategory consisting of i-torsionfree modules is extension closed for an y 1 less than or equal to i less than or equal to k if and only if every su bcategory consisting of i-th syzygy modules is extension closed for any 1 l ess than or equal to i less than or equal to k. Our results generalize the main results in Auslander and Reiten [4].