Test for Left Ideal

Theorem
Let $J$ be a subset of a ring $\struct {R, +, \circ}$.

Then $J$ is an left ideal of $\struct {R, +, \circ}$ these all hold:


 * $(1): \quad J \ne \O$


 * $(2): \quad \forall x, y \in J: x + \paren {-y} \in J$


 * $(3): \quad \forall j \in J, r \in R: r \circ j \in J$

Necessary Condition
Let $J$ be a left ideal of $\struct {R, +, \circ}$.

Then conditions $(1)$ to $(3)$ hold by virtue of the ring axioms and $J$ being a left ideal.

Sufficient Condition
Suppose conditions $(1)$ to $(3)$ hold.

Conditions $(1)$ and $(2)$ satisfy the criteria for the One-Step Subgroup Test, thus $J$ is a subgroup of $\struct {R, +}$.

As $(3)$ defines the condition for $J$ to be a left ideal, the result follows.