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If s=qr, then show the quotient ring (r)/(s) contains exactly q elements,where (r) denotes the set of the multiples of r, which I know is also the ideal of Z.

Is it possible to define a function f:(r) to something( contains q element), and use first isomorphism thm?

I think I am in the wrong way, this problem looks simple, could someone help me?

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2 Answers 2

So you really are asking to prove $\#R/Q=q$ where $R=\{nr|n\in\Bbb Z\}$ and $Q=\{nq|n\in\Bbb Z\}$. Take $f:R/Q\rightarrow \Bbb Z/q\Bbb Z, f([nr])=[n]$.

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hint: Thinking about multiplies of $r$, for every $q$ successive elements of $(r)$ there is an element of $(s)$. And so if you kill them...

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