$R$ be a ring such that every subgroup of $(R,+)$ is again an ideal of $(R,+,*)$. Then $R\cong \mathbb Z_n$.

Question

Aluffi's Algebra Chapter 0 chapter 3- section 3 problem 3.4.

Let $R$ be a ring such that every subgroup $I$ of $(R,+)$ is again an ideal of $(R,+,*)$. Then $R\cong \mathbb Z_n$ where $n$ is the characteristics of $R$.

First I guess $R$ can also be $\mathbb Z$ since it satisfies the given conditions.

My attempt was to see the hint that category of unitial rings have an initial object $\mathbb Z$ so there exists a unique map $$\phi:\mathbb Z\to R$$

Tried to observe inverse image of subgroups(since they are also ideals) but couldnot made any more sense, and also tried to use properties of being ideal.

any hint answer would be appreciated.

Answer 1

Hint: the image of $\phi$ in $R$ is a subgroup of $(R, +)$ and hence, by the given assumption, an ideal of $(R,+,\times)$ that contains $\phi(1_\Bbb{Z}) = 1_R$.