# A Prime Ideal in the Ring $\Z[\sqrt{10}]$

## Problem 223

Consider the ring
$\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}$ and its ideal
$P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.$ Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.

## Definition of a prime ideal.

An ideal $P$ of a ring $R$ is a prime ideal if whenever we have $ab \in P$ for elements $a, b \in R$, then either $a\in P$ or $b \in P$.

## Proof.

Suppose that $a+b\sqrt{10}, c+d\sqrt{10} \in \Z[\sqrt{10}]$ and the product
$(a+b\sqrt{10}) (c+d\sqrt{10}) \in P.$ Then expanding the product, we have
$ac+10bd+(ad+bc)\sqrt{10} \in P.$

Since $ac+10bd$ must be even number, we have either $a$ or $c$ is even.
Hence either
$a+b\sqrt{10}\in P \text{ or } c+d\sqrt{10} \in P,$ and we conclude that $P$ is a prime ideal.

## Further Question.

In fact, it can be proved that $P$ is a maximal ideal in the ring $\Z[\sqrt{10}]$.

For a proof, see the post “Determine the Quotient Ring $\Z[\sqrt{10}]/(2, \sqrt{10})$“.

### More from my site

• Prime Ideal is Irreducible in a Commutative Ring 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 $I$. Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is […]
• Equivalent Conditions For a Prime Ideal in a Commutative Ring Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent: (a) The ideal $P$ is a prime ideal. (b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.   Proof. […]
• Nilpotent Element a in a Ring and Unit Element $1-ab$ Let $R$ be a commutative ring with $1 \neq 0$. An element $a\in R$ is called nilpotent if $a^n=0$ for some positive integer $n$. Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.   We give two proofs. Proof 1. Since $a$ […]
• 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 […]
• 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 […]
• 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 […]
• $(x^3-y^2)$ 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 […]
• Primary Ideals, Prime Ideals, and Radical Ideals Let $R$ be a commutative ring with unity. A proper ideal $I$ of $R$ is called primary if whenever $ab \in I$ for $a, b\in R$, then either $a\in I$ or $b^n\in I$ for some positive integer $n$. (a) Prove that a prime ideal $P$ of $R$ is primary. (b) If $P$ is a prime ideal and […]

### 1 Response

1. 06/27/2017

[…] A direct proof that the ideal $P=(2, sqrt{10})$ is prime in the ring $Z[sqrt{10}]$ is given in the post “A prime ideal in the ring $Z[sqrt{10}]$“. […]

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

Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor....

Close