Ideal generated by a subset

Definition

Let $R$ be a ring and let $X \subseteq R$ be a subset. The left ideal generated by $X$ is the intersection of all the left ideals of $R$ that contain $X$. The notation for it is $(X)$. It is the smallest left ideal of $R$ that contains $X$.

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.

Explanations

L1 human

Start with some elements of the ring. The ideal that they generate contains all the sums of their multiples, and nothing else. In the integers, the multiples of $4$ and $6$ give all the even numbers, because $2 = 2 \cdot 4 - 1 \cdot 6$.