Partial orders on representations of algebras

Let k be a commutative artin ring and let Lambda be an artin k-algebra. For each natural number d let rep(d) Lambda be the set of isomorphism classes of Lambda-modules with k-length equal to d. For each natural number n, an (n x n)-matrix with entries in Lambda, can be considered as a k-endomorphism of M(n), where Mn denotes the direct sum of n copies of the Lambda-module M. The quasi-order <=(n) on rep(d) Lambda is defined by M <=(n) N if for every (n x n)-matrix p, with entries in A, we have that l(k)(M(n)/phi M(n)) <= l(k)(N(n)/phi N(n)). We show that the quasi-order <=(n) is a partial order on rep(d) Lambda for n >= d(3). (C) 2010 Elsevier Inc. All rights reserved.