Test for Ideal

Theorem
Let $J$ be a subset of a ring $\left({R, +, \circ}\right)$.

Then $J$ is an ideal of $\left({R, +, \circ}\right)$ iff these all hold:


 * $(1) \quad J \ne \varnothing$


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


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

Necessary Condition
Let $J$ be an ideal of $\left({R, +, \circ}\right)$.

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

Thus, all the conditions for $J$ being a subring hold.

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

So $J$ is an ideal of $\left({R, +, \circ}\right)$.