The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)
Problem 519
Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).
Add to solve laterProve that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).
Add to solve laterProve that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).
Add to solve later Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.
Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.
Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.
Add to solve later Prove that the ring of integers
\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.
Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.
Then prove that $R$ is a commutative ring.
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?
Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.
Prove that $R$ is a field.
Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.
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 $g-e$.
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Let $\Z$ be the ring of integers and let $R$ be a ring with unity.
Determine all the ring homomorphisms from $\Z$ to $R$.