Test for Ideal

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

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


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


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


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

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

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

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

As $r \in J \implies r \in R$, if $(3)$ holds for $J$, then $J$ is closed under $\circ$ and condition $(3)$ of Subring Test holds.

Thus, $J$ is a subring of $R$.

As $(3)$ defines the condition for $J$, being a subring, to be an ideal, the result holds.

So $J$ is an ideal of $\struct {R, +, \circ}$.