Ideal of Ring/Examples

Examples of Ideals of Rings

Set of Even Integers

The set $2 \Z$ of even integers forms an ideal of the ring of integers.

Order 2 Matrices with some Zero Entries

Let $R$ be the set of all order $2$ square matrices of the form $\begin{pmatrix} x & y \\ 0 & z \end{pmatrix}$ with $x, y, z \in \R$.

Let $S$ be the set of all order $2$ square matrices of the form $\begin{pmatrix} x & y \\ 0 & 0 \end{pmatrix}$ with $x, y \in \R$.

Then $R$ is a ring and $S$ is an ideal of $R$.