  Given a division ring K containing the field k in its center and A,B two
finite subsets of K\{0}, we give some analogues of Pl\"unnecke and Kneser
theorems for the dimension of the k-linear span of the Minkowski product AB in
terms of the dimensions of the k-linear spans of A and B. These Pl\"unnecke
type estimates are then generalized to the case of associative algebras. We
also obtain an analogue in the context of division rings of a theorem by Tao
classifying the sets of small doubling in a group.
