I know that the existence of unity implies non existence of zero divisor.
Is the converse true? (because I have prooved the latter - its quite simple).
If not; how do I go about this?
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The constant polynomial $1=1\cdot x^0$ is a unit. The ring is certainly an integral domain, being a subring of the PID $\Bbb Q[x]$. Also, $R[x]$ is an integral domain if and only if $R$ is an integral domain, see this duplicate:
Since $\Bbb Z$ is an integral domain, also $\Bbb Z[x]$ is. Also the units of $\Bbb Z[x]$ are the units of $\Bbb Z$. For this, see here: