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 $1$.

Is the Quotient Ring of an Integral Domain still an Integral Domain?
Let $R$ be an integral domain and let $I$ be an ideal of $R$.
Is the quotient ring $R/I$ an integral domain?
Definition (Integral Domain).
Let $R$ be a commutative ring.
An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]

If $ab=1$ in a Ring, then $ba=1$ when $a$ or $b$ is Not a Zero Divisor
Let $R$ be a ring with $1\neq 0$. Let $a, b\in R$ such that $ab=1$.
(a) Prove that if $a$ is not a zero divisor, then $ba=1$.
(b) Prove that if $b$ is not a zero divisor, then $ba=1$.
Definition.
An element $x\in R$ is called a zero divisor if there exists a […]

If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.
Let $R$ be a commutative ring with $1$.
Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.
Proof.
As the zero ideal $(0)$ of $R$ is a proper ideal, it is a prime ideal by assumption.
Hence $R=R/\{0\}$ is an integral […]

If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain
Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.
Definitions: zero divisor, integral domain
An element $a$ of a commutative ring $R$ is called a zero divisor […]

The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers
Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.
Prove that the quotient ring $\Z[i]/I$ is finite.
Proof.
Recall that the ring of Gaussian integers is a Euclidean Domain with respect to the norm
\[N(a+bi)=a^2+b^2\]
for $a+bi\in \Z[i]$.
In particular, […]

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 […]

Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals
Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.
Solution.
We give several examples. The key facts are:
An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain.
An ideal $I$ of […]