Test for Ideal

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

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


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

Proof

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


 * Conversely, suppose the conditions hold.

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

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

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