> [!NOTE] **Lemma** (Ideal with Unity is Whole Ring) > Let $R$ be a non-[[Zero Ring|zero]] [[Rings|ring with unity]] and let $I$ be an [[Ideal of Ring|ideal]] of $R$. Then $I=R \iff 1 \in I.$ *Proof*. ($\implies$) Suppose $I =R$. Then since $1 \in R$, $1 \in I.$ ($\impliedby$) Conversely, suppose $1 \in I$. Since $I$ is an ideal, for all $r\in R,$ $r=1 \cdot r \in I.$ Therefore $R=I.$ # Applications Corollary: [[Ideal with Unit is Whole Ring]].