Products of idempotent integer matrices

Abstract

Abstract Let E denote the set of non-identity idempotent matrices in the full matrix ring M n ( R ) over a principal ideal domain R . A necessary and sufficient condition is found for the subsemigroup generated by E to be the set of all matrices in M n ( R ) of rank less than n . The condition is satisfied when R is a discrete valuation ring and when R is the ring of integers. Thus every n × n matrix of rank less than n is a product of idempotent integer matrices.