# Definition of principal ideal

This is a pretty basic question about principal ideals - on page 197 of Katznelson's A (Terse) Introduction to Linear Algebra, it says:

Assume that $\mathcal{R}$ has an identity element. For $g\in \mathcal{R}$, the set $I_g = \{ag:a\in\mathcal{R}\}$ is a left ideal in $\mathcal{R}$, and is clearly the smallest (left) ideal that contains $g$.

Ideals of the form $I_g$ are called principal left ideals...

(Note $\mathcal{R}$ is a ring).

Why is the assumption that $\mathcal{R}$ has an identity element important?

Because if $\mathcal{R}$ has an identity, then $I_{g}$ is the smallest left ideal containing $g$. Without an identity, it might be that $g \notin I_{g}$.
For instance if $\mathcal{R} = 2 \mathbf{Z}$, then $I_{2} =\{a \cdot 2:a\in 2 \mathbf{Z} \} = 4 \mathbf{Z}$ does not contain $2$.
If $R$ is a rng, the left ideal generated by an element $g$ (i.e. the smallest left ideal containing $g$) is given by the $\mathbb{Z}$-span of $\{g\} \cup \{rg : r \in R\}$. Thus, elements of $\langle g \rangle$ have the form $z \cdot g + r \cdot g$ for $z \in \mathbb{Z}$ and $r \in R$.