> [!NOTE] Theorem (Intersection of Ring Ideals is Ideal) > Let $R$ be a [[Rings|ring]]. Let $I,J \subset R$ be [[Ideal of Ring|ideals]] of $R.$ Then their intersection $I \cap J$ is an ideal of $R.$ *Proof*. (Identity) $0 \in I+J$ since $0\in I$ and $0\in I$ as $I,J$ are subgroups of $R.$ (Closure) Let $x,y\in I\cap J.$ Then $x,y\in I$ and $x,y\in J.$ Since $I$ and $J$ are subgroups, they are closed under addition: $x+y \in I$ and $x+y \in J.$ So $x+y \in I \cap J.$ (Inverses) $-x\in I$ and $-x\in J$ so $-x \in I\cap J.$ Thus by [[Two-Step Subgroup Test]], $I\cap J$ is also a subgroup of $R.$ Now let $r\in R.$ Then $r\times x \in I$ and $r\times x \in J$ so $r\times x \in I \cap J.$ Similarly $x \times r \in I \cap J.$ Therefore $I \cap J$ is also an ideal of $R.$