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 $$\left({R, +, \circ}\right)$$.