Ideal generated by a subset

Remarks

The left ideal $(X)$ is the set of the finite sums $r_1 x_1 + \dots + r_n x_n$ with $n \ge 0$, $r_i \in R$ and $x_i \in X$. For a finite set, one writes $(x_1, \dots, x_n)$ for $(\{x_1, \dots, x_n\})$. In $\mathbb{Z}$, the ideal $(4, 6)$ is $2\mathbb{Z}$. An ideal generated by one element is a principal ideal.