Let $R$ be a commutative ring and let $a\in R$.
Show that $I=\{x\in R\mid ax=0\}$ is an ideal.
For all $b \in R$, $$bI=b\{x\in R\mid ax=0\}=\{bx\in R\mid a(bx)=0\} =\{xb\in R\mid a(x)=0\}=Ib\;,$$ thus I can say $I$ is an ideal of $R$, right? Thanks