Test for Left Ideal
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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.
$\blacksquare$