Let $R=2\mathbb{Z}$, then $R$ is a ring without identity, and we consider the ideal generated by $4$, that is $$(4)=\{4(2j)+4\overline{k}\;|\;j,\overline{k}\in\mathbb{Z}\}.$$
I must prove that $(4)=4\mathbb{Z}$. Now, I proved that $(4)\subseteq4\mathbb{Z}$, in fact, let $x\in(4)$, then $x=4(2j)+4\overline{k}=4(2j+\overline{k})\in 4\mathbb{Z}$.
Can I prove that $4\mathbb{Z}\subseteq (4)$? Thanks!
Let $x$ be a multiple of $4$, so $x=4k$, with integer $k$. Then $$x=4k=4(2\cdot0)+4k\in(4)$$