Test for Left Ideal

Theorem

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

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

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

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

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

Proof

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.

$\Box$

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.

$\blacksquare$