## 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\}.\]

**(a)** True or False: $S$ is a subring of $R$.

**(b)** True or False: $S$ is an ideal of $R$.