can there exists subring of nonzero characteristic if ring has zero characteristic?

Question

Question Let $R$ be a ring such that $\operatorname{char}(R)$ is $0$ and $S$ is subring of $R$ then $\operatorname{char}(S)$ is $0$.

My attempt: If $\operatorname{char}(R)=0$ then $\nexists n\in\mathbb{N}$ such that, $nx=0$ for all $x\in R$. I don't know how to go proceed further. please help...

Note: ring $R$ may not have unity...

Answer 1

If by "ring" you mean "ring with unity", and subrings are therefore required to contain the unity, then it is easy to see that the answer is "no", since in that setting the characteristic is the nonnegative generator of the kernel of the ring map $\mathbb{Z}\to R$ given by sending $1$ to $1_R$.

If you either do not require rings to have a unity, or do not require subrings to have the same unity as their overrings (uncommon when you do require a unity), then taking the direct product of a ring of characteristic $0$, $R_0$, and a ring of positive characteristic $R_c$, will give you an example $R_0\times R_c$: because the product has characteristic $0$, but $\{0\}\times R_c$ has characteristic $\mathrm{char}(R_c)\gt 0$.

Answer 2

Subrings are closed under addition and the inclusion of $S$ into $R$ is an injection, so if $n\cdot1=0$ in $S$ then that is also the case in $R$.