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If R is a commutative ring containing a field K as subring and assume that R is finite dimensional vector space over K then show that every prime ideal of R will be a maximal ideal.
My approach was to show for any prime ideal I, R/I will be a finite integral domain and hence a field which will result in I being maximal. But, here it is not given that K is finite field. Since I have just started studying this course and don't know much about Artinian or Noetherian ring so can anyone please solve this problem for infinite field using very basic ideas of ring & vector space theory and not going into domain of Artinian or Noetherian ring.