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Given a division ring K containing the field k in its center and A,B twofinite subsets of K\{0}, we give some analogues of Pl\"unnecke and Knesertheorems for the dimension of the k-linear span of the Minkowski product AB interms of the dimensions of the k-linear spans of A and B. These Pl\"unnecketype estimates are then generalized to the case of associative algebras. Wealso obtain an analogue in the context of division rings of a theorem by Taoclassifying the sets of small doubling in a group.

Pl\"unnecke and Kneser type theorems for dimension estimates