Minimal underlying division rings of sets of points of a projective space

Let V be a vector space over a division ring K. Let P be a spanning set of points in Sigma := PG(V). Denote by K(P) the family of sub-division rings F of K having the property that there exists a basis B-F of V such that all points of P are represented as F-linear combinations of BF. We prove that when K is commutative, then K (P) admits a least element. When K is not commutative, then, in general, K (P) does not admit a minimal element. However we prove that under certain very mild conditions on P, any two minimal elements of K (P) are conjugate in K, and if K is a quaternion division algebra then K (P) admits a minimal element. (c) 2007 Elsevier Inc. All rights reserved.