# Idempotent Elements and Zero Divisors in a Ring and in an Integral Domain

## Problem 516

Prove the following statements.

**(a)** If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.

**(b)** Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.

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## Definitions (Idempotent, Zero Divisor, Integral Domain)

Let $R$ be a ring with $1$.

- An element $a$ of $R$ is called
**idempotent**if $a^2=a$. - An element $a$ of $R$ is called
**zero divisor**if there exists a nonzero element $x$ of $R$ such that $ax=0$ or $xa=0$. - A commutative ring that does not have a nonzero zero divisor is called an
**integral domain**

## Proof.

### (a) If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.

By definition of an idempotent element, we have $a^2=a$.

It yields that

\begin{align*}

a(a-1)=a^2-a=0.

\end{align*}

Since $a\neq 1$, the element $a-1$ is a nonzero element in the ring $R$.

Thus $a$ is a zero divisor.

### (b) Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.

Suppose that $a$ is an idempotent element in the integral domain $R$.

Thus, we have $a^2=a$.

It follows that we have

\begin{align*}

a(a-1)=a^2-a=0. \tag{*}

\end{align*}

Since $R$ is an integral domain, there is no nonzero zero divisor.

Hence (*) yields that $a=0$ or $a-1=0$.

Clearly, the elements $0$ and $1$ are idempotent.

Thus, the idempotent elements in the integral domain $R$ must be $0$ and $1$.

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