Is the Given Subset of The Ring of Integer Matrices an Ideal?
Problem 524
Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients:
\[R=\left\{\, \begin{bmatrix}
a & b\\
c& d
\end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.\]
Let $S$ be the subset of $R$ given by
\[S=\left\{\, \begin{bmatrix}
s & 0\\
0& s
\end{bmatrix} \quad \middle | \quad s\in \Z \,\right\}.\]
In fact, let
\[A= \begin{bmatrix}
t & 0\\
0& t
\end{bmatrix}\text{ and } B=\begin{bmatrix}
s & 0\\
0& s
\end{bmatrix}\]
be arbitrary elements in $S$ with $t, s\in \Z$.
Then we have
\begin{align*}
A+B=\begin{bmatrix}
t+s & 0\\
0& t+s
\end{bmatrix} \in S
\end{align*}
and
\begin{align*}
AB=\begin{bmatrix}
ts & 0\\
0& ts
\end{bmatrix} \in S.
\end{align*}
Hence $S$ is closed under addition and multiplication.
Note that the $2\times 2$ identity matrix is the unity element of $R$ as well as the unity element of $S$.
Thus, the subset $S$ is a subring of $R$.
(b) True or False: $S$ is an ideal of $R$.
False.
To see that $S$ is not an ideal of $R$, consider the element
\[\begin{bmatrix}
1 & 1\\
1& 1
\end{bmatrix} \in R\]
and the element
\[\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix} \in S.\]
Then we have
\begin{align*}
\begin{bmatrix}
1 & 1\\
1& 1
\end{bmatrix}\begin{bmatrix}
1 & 0\\
0& 1
\end{bmatrix}=\begin{bmatrix}
1 & 1\\
1& 1
\end{bmatrix},
\end{align*}
which is not in $S$.
This implies that $S$ is not an ideal of $R$.
(If $S$ were an ideal of $R$, then an element of $S$ multiplied by an element of $R$ would stay in $S$.)
$(x^3-y^2)$ 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 $R[t]$.
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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
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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.
Proof.
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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.
Hint.
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Let $R$ be a commutative ring with $1$.
Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.
Proof.
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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
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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.
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