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.

An element $a$ of a commutative ring $R$ is called a zero divisor if there is $b\neq0$ in $R$ such that $ab=0$.

If a ring $R$ contains no nonzero zero divisors, then we call $R$ an integral domain.

Proof.

Suppose that we have
\[ab=0\]
for $a, b \in R$. To show that $R$ has no nonzero zero divisors, we need to prove that $a$ or $b$ is the zero element.
Since $ab=0\in P$ and $P$ is a prime ideal, either $a\in P$ or $b\in P$.
Without loss of generality, we may assume $a\in P$.

If $a=0$, then we are done.
So assume that $a\neq 0$. Then since $P$ does not contain any nonzero zero divisor, we must have $b=0$, otherwise $ab=0, b\neq 0$ means that $a$ is a nonzero zero divisor in $P$.
Therefore, in any case we have either $a=0$ or $b=0$, and thus the ring $R$ contains no nonzero zero divisors. Hence $R$ is an integral domain.

Characteristic of an Integral Domain is 0 or a Prime Number
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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 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.
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$(x^3-y^2)$ 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 […]

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Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer.
Prove that the polynomial
\[f(x)=x^n-t\]
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Proof.
Consider the principal ideal $(t)$ generated by $t$ […]

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Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent:
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Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be irreducible if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$.
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