Question about dimension of a vector space

I'm reading Exercises in Classical Ring Theory (T.Y.Lam) and there is a exercise:

I'm not sure how to determine the left (right) dimension of the vector spaces (red underline in the above).My though is that:

• For $[K\cdot x : K]_r$ : the basis includes two vectors : $(xK; 0)$ and $(txK; 0)$.
• For $[D:K]_l$ : I can not determine the basis.

My question is: How can you determine the dimension of the vector spaces that are underlined?