A commutative ring $R$ is a principal ideal domain (PID) if $R$ is a domain and any ideal $I$ is generated by a single element $a\in I$, that is $I=(a)$.

Proof.

Since $R$ is a PID, we can write $P=(a)$, an ideal generated by an element $a\in R$.
Since $P$ is a nonzero ideal, the element $a\neq 0$.

Now suppose that we have
\[P \subset I \subset R\]
for some ideal $I$ of $R$.

We can write $I=(b)$ for some $b \in R$ since $R$ is a PID.
The element $a\in (a) \subset (b)$ and so there is an element $c \in R$ such that $a=bc$.

Since $a=bc$ is in the prime ideal $P$, we have either $b \in P$ or $c \in P$.
If $b\in P$, then it follows that $I=(b)\subset P$, and hence $P=I$.
If $c \in P=(a)$, then we have $d\in R$ such that $c=ad$.

Then we have
\begin{align*}
a=bc=bad
\end{align*}
and since $R$ is a domain and $a\neq 0$, we have
\[1=bd.\]
This yields that $b$ is a unit and hence $I=(b)=R$.

In summary, we observe that whenever we have $P \subset I \subset R$, we have either $I=P$ or $I=R$. Thus $P$ is a maximal ideal.

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 […]

Every Maximal Ideal of a Commutative Ring is a Prime Ideal
Let $R$ be a commutative ring with unity.
Then show that every maximal ideal of $R$ is a prime ideal.
We give two proofs.
Proof 1.
The first proof uses the following facts.
Fact 1. An ideal $I$ of $R$ is a prime ideal if and only if $R/I$ is an integral […]

Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID
(a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.
(b) Prove that a quotient ring of a PID by a prime ideal is a PID.
Proof.
(a) Prove that every PID is a maximal ideal.
Let $R$ be a Principal Ideal Domain (PID) and let $P$ […]

$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.
Proof.
Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$.
Define the function $\Psi:R[x,y] \to R[t]$ sending […]

The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal
Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element.
Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.
Proof.
Suppose that we have an ideal $I$ of $R$ such that
\[(a) \subset I \subset […]

Every Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative Ring
Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.
Then prove that every prime ideal is a maximal ideal.
Hint.
Let $R$ be a commutative ring with $1$ and $I$ be an ideal […]

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 […]