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Question about dimension of a vector space

I'm reading Exercises in Classical Ring Theory (T.Y.Lam) and there is a exercise:Image may be NSFW.
Clik here to view.
enter image description here

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?


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