For a prime p and a matrix AβZnΓn, write A as A=p(Aquop)+(Aremp) where the remainder and
quotient operations are applied element-wise. Write the p-adic expansion of
A as A=A[0]+pA[1]+p2A[2]+β― where each A[i]βZnΓn has entries between [0,pβ1]. Upper bounds are
proven for the Z-ranks of Aremp, and Aquop. Also, upper bounds are proven for the
Z/pZ-rank of A[i] for all iβ₯0 when p=2, and
a conjecture is presented for odd primes.Comment: 8 page