PRODUCTS OF IDEMPOTENT INTEGER MATRICES

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 x n matrix of rank less than n is a product of idempotent integer matrices.