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
(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
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
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 $1$.
Characteristic of an Integral Domain is 0 or a Prime Number
Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
Definition of the characteristic of a ring.
The characteristic of a commutative ring $R$ with $1$ is defined as […]