Let ∫, g : E → E, h : E → F, k : F → G be linear transformations of left vector spaces over a division ring D with dim_DE = n, dim_DF = m, dim_DG = p. 

(a) Rank (∫ + g) ≤ rank ∫ + rank g.

(b) Rank (kh) ≤ min {rank h, rank k}.

(c) Nullity kh ≤ nullity h + nullity k.

(d) Rank ∫ + rank g - n ≤ rank ∫g ≤ min {rank ∫, rank g}. 

(e) Max {nullity g, nullity h} ≤ nullity hg.

(f) If m ≠ n, then (e) is false for h and k.