Subgroups and ideals of integer numbers. Let $(\mathbb{Z},+)$ be the additive group of integers, and $(\mathbb{Z}, +, \cdot)$ the ring of integers. By definition, every ideal of $(\mathbb{Z}, +, \cdot)$ is a subgroup of $(\mathbb{Z},+)$. Is the opposite true? Is every subgroup of $(\mathbb{Z},+)$ also an ideal of $(\mathbb{Z}, +, \cdot)$? If yes, how can it be proved? If not, is there any counterexample? 

Every subgroup of $(\mathbb{Z},+)$ is of the form $(n\mathbb{Z},+)$ with $n\in\mathbb{N}$. A proof uses division with remainder. I wrote it down here: Subgroups of $\mathbb{Z}$
