Rings where every subgroup of the additive group is an ideal?

Question

The rings $\mathbb Z$ and $\mathbb Z/n\mathbb Z$ have the property that every subgroup of the additive group is also an ideal (i.e., every subgroup absorbs multiplication by all ring elements). This is because every element of these rings can be written as the sum of $1$'s, so multiplication is the same as repeated addition. Are there any other rings for which this is true?

Answer 1

For such a ring with identity we must have that the subgroup generated by $1$ is always an ideal. This means that for any element $x$ we have that $x\cdot 1$ is a finite sum of $1$'s. This shows that the example you give is the only such example as there is a unique ring homomorphism $\mathbb{Z}\to R$ sending $1\mapsto 1$, which would be surjective for such a ring.