# Definition:Ideal of Ring/Left Ideal

## Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$.

$J$ is a left ideal of $R$ if and only if:

$\forall j \in J: \forall r \in R: r \circ j \in J$

that is, if and only if:

$\forall r \in R: r \circ J \subseteq J$