Problem about ring and integral domain

Question

Assume we have ring $\langle \{5x|x \in I \}, +,\cdot \rangle$, $I$ is integers set, $+$ and $\cdot $ is common plus and times.

The ring is not integral domain, because it doesn't contain multiplication identity element.

Why?

Answer 1

Because an identity is a nonzero element satisfying $1^2=1$, and there are no such elements in that rng.

In general, in any domain without zero divisors and with identity, the equation $x^2=x$ can only have two solutions since $x(x-1)=0$ implies $x=0$ or $x=1$. Your subring can't provide any more solutions than $\Bbb Z$ already does, being a subset of $\Bbb Z$.