Let \(R\) be a commutative ring. An ideal \(I\normal R\) is a subset \(I\subseteq R\) such that \((I, +)\) is a subgroup of \((R, +)\) and \(ri \in I\) for all \(r\in R\) and \(i\in I\).
Let \(R\) be a commutative ring. A nonempty subset \(I\subseteq R\) is an ideal if and only if \(I + I \subseteq I\) and \(RI \subseteq I\).