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To prove the statement:Any finite integral domain $R$ is a field. Let $r \in R$ , we define a function$ f: R \rightarrow R$ by $f(x)=rx$. The proof says: this map is injective(i understand this), then since $R$ is finite, this map is also surjective(why?)

How does it follow from "$R$ is finite" that $f$ is surjective?

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If $f$ is injective, $f(R)$ has the same cardinality as $R$.

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  • $\begingroup$ oh yes!!! you are right! thank you $\endgroup$ – bbw Nov 21 '17 at 19:37
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You can use a proof by contradiction.

Assume it is injective but not surjective--that is, that there exists an element of R that is not in the image $f(R)$.

Since there is at least one element of $R$ that is not in the image of $R$, the size of the image $f(R)$ is at most $|R|-1$. So our mapping $f$ maps $|R|$ elements onto an image set of size $|R|-1$, and by the pigeonhole principle, at least one element of $f(R)$ must be the image of 2 distinct elements of $R$. This contradicts our assumption of injectivity.

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Here's another (more roundabout) way to get a contradiction: Divide $R$ in to two disjoint subsets: $A$, the set of elements of $R$ that are in $f(R)$, and $B$, the set of elements of $R$ that are not in $f(R)$. So $A$ and $B$ are disjoint, and we have $R=A \cup B$. Now, \begin{align}|R|&=|A|+|B|-\underbrace{|A \cap B|}_{\substack{=0}}\\ &=|A|+|B|\\ &> |A|, \end{align} since we assumed that $f$ was not surjective, and, hence, $B$ is nonempty. But $f$ is injective, so $|A|=|f(R)|=|R|$, and this gives us $$|R|>|R|.$$

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