Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals

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• Every Prime Ideal of a Finite Commutative Ring is Maximal Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$. Proof. We give two proofs. The first proof uses a result of a previous problem. The second proof is self-contained. Proof 1. Let $I$ be a prime ideal […]
• The Ideal $(x)$ is Prime in the Polynomial Ring $R[x]$ if and only if the Ring $R$ is an Integral Domain Let $R$ 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 […]
• Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$ Let \[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\] be an ideal of the ring \[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\] Then determine the quotient ring $\Z[\sqrt{10}]/P$. Is $P$ a prime ideal? Is $P$ a maximal ideal?   Solution. We […]
• 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 […]
• 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 […]
• A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring Let $R$ be the ring of all continuous functions on the interval $[0, 2]$. Let $I$ be the subset of $R$ defined by \[I:=\{ f(x) \in R \mid f(1)=0\}.\] Then prove that $I$ is an ideal of the ring $R$. Moreover, show that $I$ is maximal and determine […]
• 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 […]
• In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$. Show that $P$ is a maximal ideal in $R$.   Definition 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 […]