Find the dimension of
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if by "dimension" you mean dimension as a vector space over , the answer is . this is easy to see: put and and let be the ideal of which is generated by and . show that and conclude that is an -basis for your ring.
I get everything up to the last conclusion: why does imply that is a basis?
and so every element of your ring is in the form , for some also, since , we have , for some so generates your ring. it is obvious that the set is -linearly independent.
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