**Definition 11**. Let \(R\) be a commutative ring and \(\emptyset \ne I \subseteq R\). Then
\(I\) is an **ideal** of \(R\) if:

- The sum \(a + b\) is in \(I\) for all \(a, b \in I\).
- The product \(ar\) is in \(I\) for all \(a \in I\) and \(r \in R\).

The ideal \(I\) is **proper** if \(I \ne R\).

**Remark 12**. Definition 11 says that if \(R\) is a commutative ring,
and \(\emptyset \neq I \subseteq R\), then \(I\) is an ideal of \(R\) if:

- The set \(I\) is closed under addition.
- The set \(I\) is closed under multiplication by any \(r \in R\).

**Proposition 13**. Let \(I\) be an ideal of a commutative ring \(R\).
Then:

- The equation \(a \cdot 0 = 0 \cdot a = 0\) holds for all \(a \in R\).
- The element \(0\) is in \(I\).
- If \(1 \in I\) then \(I\) is not proper.