# Category: Ring theory

## Problem 345

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 $R/I$.

## Problem 333

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
$f(x)=x^n-t$ in the ring $S[x]$ is irreducible in $S[x]$.

## Problem 318

Let $R$ be a ring with unity.
Suppose that $f$ and $g$ are ring homomorphisms from $\Q$ to $R$ such that $f(n)=g(n)$ for any integer $n$.

Then prove that $f=g$.

## Problem 302

Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by
$\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,$ where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the augmentation map and the kernel of $\epsilon$ is called the augmentation ideal.

(a) Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.

(b) Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.

## Problem 264

Let $\Z$ be the ring of integers and let $R$ be a ring with unity.
Determine all the ring homomorphisms from $\Z$ to $R$.

## Problem 247

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 $a^n\in P$ for some $a\in R$ and a positive integer $n$, then show that $a\in P$.

(c) If $P$ is a prime ideal, prove that $\sqrt{P}=P$.

(d) If $Q$ is a primary ideal, prove that the radical ideal $\sqrt{Q}$ is a prime ideal.

## Problem 239

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

## Problem 234

Show that the polynomial
$f(x)=x^4-2x-1$ is irreducible over the field of rational numbers $\Q$.

## Problem 228

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 $p$.

## Problem 224

In the ring
$\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},$ show that $5$ is a prime element but $7$ is not a prime element.

## 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 $\Z[\sqrt{10}]$.

## 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.

## Problem 204

Is there a (not necessarily commutative) ring $R$ with $1$ such that the equation
$x+x=1$ has more than one solutions $x\in R$?

## Problem 203

Let $R$ be a commutative ring. Let $S$ be a subset of $R$ and let $I$ be an ideal of $I$.
We define the subset
$(I:S):=\{ a \in R \mid aS\subset I\}.$ Prove that $(I:S)$ is an ideal of $R$. This ideal is called the ideal quotient, or colon ideal.

## Problem 199

Let $R$ be the ring of all continuous functions on the interval $[0,1]$.
Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$.

Show that the set $I$ is an ideal of $R$ but is not a prime ideal.

## Problem 198

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 field.

## Problem 197

Let $R$ be a ring with unit $1\neq 0$.

Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$.
(Do not assume that the ring $R$ is commutative.)

## Problem 192

Show that any finite integral domain $R$ is a field.

## Problem 188

Denote by $i$ the square root of $-1$.
Let
$R=\Z[i]=\{a+ib \mid a, b \in \Z \}$ be the ring of Gaussian integers.
We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to
$N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.$

Here $\bar{\alpha}$ is the complex conjugate of $\alpha$.
Then show that an element $\alpha \in R$ is a unit if and only if the norm $N(\alpha)=\pm 1$.
Also, determine all the units of the ring $R=\Z[i]$ of Gaussian integers.

## Problem 179

Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.