Question

Prove that the only idempotents in an integral domain R are ${\mathbf{0}}_{\mathbf{R}}$and ${\mathbf{1}}_{\mathbf{R}}$. (See Exercise 3.)

Short answer

It is proved that there are only two idempotent $0_R$ and$1_R$ in the integral domainR.

Step-by-step solution

To prove

Here, we need to prove that for any integral domain R, there are only two idempotent $0_R$and $1_R$.

Proof

Let the idempotent element, $b\in R$, such that, $bb=b$. Then, we have:

$\begin{array}{rcl}bb& =& b\\ bb-b& =& 0_R\\ bb+(-1_R)b& =& 0_R\\ (b-1_R)b& =& 0_R\\ & \Rightarrow & b-1_R=0_R\mathrm{or}b=0_R\end{array}$

Which again implies that, $b=1_R$or $b=0_R$.

Hence, there are only two idempotentrole="math" localid="1648212886682" $0_R$ and$1_R$ in the integral domain R.