ring contains a subring isomorphic to $Z$ implies it is integral domain?

Question

If a ring contains a subring isomorphic to $Z$,does it necessarily imply that the ring must be an integral domain?

I know that it must have characteristic 0.However I cannot proceed further.

Answer 1

In general, no. Take $\mathbb{Z}\times R$, where $R$ is a unit ring. It is not an integral domain, since $(1,0).(0,1)=(0,0)=0$. But it contains a subring isomorphic to the integers.

Answer 2

No; consider the ring $\mathbb Z[\epsilon]/(\epsilon^2)$. That is, the ring consisting of elements of the form $$ a + b\epsilon\qquad a,b\in\mathbb Z, $$ with obvious addition and multiplication given by $$ (a + b\epsilon)(c + d\epsilon) = ac + (ad + bc)\epsilon. $$ Then $\epsilon^2 = 0$, even though $\epsilon \neq 0$.