On idempotent factorizations in the endomorphism ring of a free module of finite rank

Abstract Factorization Theorems with Applications to Idempotent Factorizations

Let be a preorder on a monoid and be an integer the of an is the sup of the integers for which there is a (strictly) sequence of of with (with where is a if and a otherwise.we establish that, if is then each factors through the of degree where a of degree is a that cannot be written as a product of or fewer each of which is (strictly) smaller than with respect to.In addition, we show that, if is strongly then factors through the of where a is a in the process, we also obtain upper bounds for the length of a shortest factorization of (into either of degree or in terms of its).We specialize these abstract results to the case in which is the multiplicative submonoid of a ring formed by the zero divisors and theidentity and is the preorder on defined by iff where denotes a rightannihilator.