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) $$J \ne \varnothing$$;
 * 2) $$\forall x, y \in J: x + \left({-y}\right) \in J$$;
 * 3) $$\forall j \in J, r \in R: x \circ j \in J, j \circ r \in J$$.

Proof

 * If $$J$$ is an ideal of $$\left({R, +, \circ}\right)$$, the conditions hold by virtue of the ring axioms and $$J$$ being an ideal.


 * Conversely, suppose the conditions hold.

As $$r \in R \Longrightarrow r \in J$$, 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.