> [!NOTE] Lemma (Ideal with Unit 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 $I$ contains a [[Unit in a Ring|unit]] in $R$. > **Proof.** ($\implies$) Suppose $I=R$. Then, since $1 \in R$, $1 \in I$ and $1$ is a unit. ($\impliedby$) Conversely, suppose $u \in I$ and $u$ is a unit. This means that there exists $v \in R$ such that $uv=vu=1$. By the 'multiple property' of the ideal $I$, $1= uv \in I$ and $I=R$ by [[Ideal with Unity is Whole Ring]].