Prove that every finite domain contains an identity element.
Please give me help
Prove that every finite domain contains an identity element.
Please give me help
Let $a$ be a non-zero element of the domain. Consider the objects $a$, $a^2$, $a^3$, and so on.
The powers of $a$ cannot all be different, since if they were, there would be infinitely many elements in the domain. It follows that there are natural numbers $m$ and $n$, with $m
Hint $\:$ The elements $\ne 0$ form a nonempty finite semigroup, so contain an idempotent $\rm\:e.\:$ Thus $\rm\: 0\: \ne\: e\: =\: e^2\: \ \Rightarrow\ \ e\:x\: =\: e^2\:x\ \ \Rightarrow\ \ x\: =\: e\:x$
Alternatively, notice that $\rm\ a\ne 0\:\Rightarrow\: x\mapsto a\:x\ $ is $1$-$1$ so onto, therefore
$\qquad\qquad$ for all $\rm\:x\!:\ $ $\rm\begin{eqnarray}\exists\: e\!:\ \ a\: &=&\:\rm \color{#C00}{a\:e}\\ \rm \exists\: d\!:\ \ x\: &=&\:\rm a\:d\end{eqnarray}$ $\ \ \Rightarrow \ \ \begin{eqnarray}\rm a\:d \: &=&\: \rm \color{#C00}{e\:a}\:d\\ \rm x \:&=&\: \rm e\:x\end{eqnarray} $
The proof in André's answer is a special case of the first method above. Probably the proof of Herstein mentioned by Chandrasekhar is similar to the second proof above.
This is a property that every finite integral domain $D$ is a field. Please look into I.N.Herstein's text.
Idea. Is to consider a non-zero element in $D$ and establish a bijection from $D \to D$ via $x \mapsto ax$.
Please look into Page 128 of the following link