Let \(R\) be a commutative ring and let \(S\subseteq R\) be a subset. The ideal generated by \(S\) is \[ \gens{S} \coloneqq \theset{ \sum_{i=1}^n r_i s_i \st n\geq 0,\ r_i\in R,\ s_i\in S }. \]
For \(a, b\in R\), \[ \gens{a, b} = Ra + Rb = \theset{ r_1 a + r_2 b \st r_1, r_2 \in R }. \]