## If $R$ is a Noetherian Ring and $f:R\to R’$ is a Surjective Homomorphism, then $R’$ is Noetherian

## Problem 413

Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

of the day

Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

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 $R$.

Add to solve laterLet $f:R\to R’$ 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 $R$.

Add to solve later For each positive integer $n$, prove that the polynomial

\[(x-1)(x-2)\cdots (x-n)-1\]
is irreducible over the ring of integers $\Z$.

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.

Add to solve laterLet $R$ be a commutative ring with unity.

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

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 $R/I$.

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 $S[x]$.

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 $f=g$.

Add to solve laterLet $R$ 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 $g-e$.

Read solution

Let $\Z$ be the ring of integers and let $R$ be a ring with unity.

Determine all the ring homomorphisms from $\Z$ to $R$.

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.

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 laterShow that the polynomial

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

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 $p$.

Add to solve later 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.

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

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.

Add to solve laterIs there a (not necessarily commutative) ring $R$ with $1$ such that the equation

\[x+x=1 \]
has more than one solutions $x\in R$?

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