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Let $f: R\to S$ be a homomorphism of rings (with $R$ commutative) such that kernel of $f$ has $4$ elements and image of $f$ has $16$ elements. How many elements does $R$ have?

Would you simply use the first isomorphism theorem in the following way? $R/\ker(f)=R/4$, so $R$ must have $16\cdot 4= 64$ elements.

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    The correct calculation looks like that. But you have details wrong. e.g. the assertion $R/\ker(f) = R/4$ is way off base.2012-03-31
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    In more proper notation $16=|{\rm Im}(f)|=|R/\ker(f)|=|R|/|\ker(f)|=|R|/4$ so that $|R|=16\cdot4=64$.2012-03-31

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