# Category: Ring theory

## Problem 487

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?

## Problem 437

Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.
Prove that $R$ is a field.

## Problem 436

Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.

1. The ring $R$ is a field.
2. The only ideals of $R$ are $(0)$ and $R$.
3. Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.

## Problem 413

Suppose that $f:R\to R’$ is a surjective ring homomorphism.
Prove that if $R$ is a Noetherian ring, then so is $R’$.

## Problem 412

Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.

Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.

## Problem 411

Let $f:R\to R’$ be a ring homomorphism. Let $I’$ be an ideal of $R’$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.

## Problem 372

For each positive integer $n$, prove that the polynomial
$(x-1)(x-2)\cdots (x-n)-1$ is irreducible over the ring of integers $\Z$.

## Problem 360

Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have
$I_1+I_2=R.$

Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.

## Problem 351

Let $R$ be a commutative ring with unity.
Then show that every maximal ideal of $R$ is a prime ideal.

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