How fo show if $(R,+)$ is cyclic group then is commutative

Question

Show that $(R,+)$ is cyclic group, then $R$ is a commutative ring.

I know if $R$ is cyclic then $ R= \left<a\right>= \{a^n , n \in \Bbb Z\}$.

I do not have any idea for solving problem.can any one prove this problem

Thanks in advance

Answer 1

Hint. Show that the additive group underlying $R$ is cyclic if and only if it is generated by $1$ and then consider the unique ring homomorphism $ℤ → R,~1 ↦ 1$.

If $a$ is a generator of $⟨R, +, 0⟩$, then $1 = na = an$ for some $n ∈ ℤ$. What does this tell you about $a$? What does this tell you about the additive group homomorphism $R → R,~x ↦ ax$?