If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain

Prime Ideal Problems and Solution in Ring Theory in Mathematics

Problem 220

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.

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Definitions: zero divisor, 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.


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.

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