## Every Maximal Ideal of a Commutative Ring is a Prime Ideal

## Problem 351

Let $R$ be a commutative ring with unity.

Then show that every maximal ideal of $R$ is a prime ideal.

Let $R$ be a commutative ring with unity.

Then show that every maximal ideal of $R$ is a prime ideal.

Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.

Add to solve laterLet $R$ 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 $p$.

Add to solve laterLet $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.

Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.

Add to solve laterLet $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.

Show that $P$ is a maximal ideal in $R$.

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