initial ramification index of noninvariant valuations on finite dimensional division algebras

let d be a division ring with centre k and dim, d< ? a valuation on k and v a noninvariant extension of ? to d. we define the initial ramfication index of v over ?, ?(v/ ?) .let a be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to d with valuation rings a , a ,…, a . if b= a , it is shown that the following conditions are equivalent: (i) b is a finite a-module, (ii) b is a free a-module, (iii) [b/mb: a/m] = [d: k], (iv) e(v / ?) f(v / ?)= [d: k] and ? (v / ?)= e(v / ?). it is also proved that if ? (v/ ?) = e(v/ ?), and any of (i) - (iv) holds, then v is invariant