[SOLVED] Vector spaces over division rings.

Hi:

I have the following definition: let V be a vector space over

a division ring D. A mapping a of V into V is called a linear trans-

formation of V if it has the followiwng two properties:

(x+y)a = xa+ya for x,y $\displaystyle \in$ V,

(x$\displaystyle \alpha$)a = (xa)$\displaystyle \alpha$ for x $\displaystyle \in$ V, $\displaystyle \alpha$ D

And here I find an odd thing. If a is the mapping multiplication by

a scalar (that is, by an element of D), then a is not in general a linear trans-

formation of V according to the definition, because D needs not be

commutative. Any hint will be welcome.