Principal ideal

Definition

Let $R$ be a ring. A left ideal $I$ of $R$ is principal if there is an element $a \in R$ such that $I = (a)$, the left ideal generated by $\{a\}$.

Remarks

The left ideal generated by $a$ is $R a = \{r a : r \in R\}$. If $R$ is commutative, the principal ideal $(a)$ is the set of the multiples of $a$. Every ideal of $\mathbb{Z}$ is principal: each one is $n\mathbb{Z}$ for some integer $n \ge 0$. In $\mathbb{Z}[x]$, the ideal $(2, x)$ is not principal.

Explanations

L1 human

A principal ideal is the set of all the multiples of one element. For example, the multiples of $3$ in the integers form the principal ideal $(3)$.