Ranks of Quotients, Remainders and p-Adic Digits of Matrices

For a prime p and a matrix A in Z^{n times n}, write A as A = p (A ,quo, p) + (A ,rem, p) where the remainder and quotient operations are applied element-wise. Write the p-adic expansion of A as A = A^{[0]} + p A^{[1]} + p^2 A^{[2]} + cdots where each A^{[i]} in Z^{n times n} has entries between [0, p-1]. Upper bounds are proven for the Z-ranks of A ,rem, p, and A ,quo, p. Also, upper bounds are proven for the Z/pZ-rank of A^{[i]} for all i ge 0 when p = 2, and a conjecture is presented for odd primes.