I am in the beginning of studing of ring theory so let me ask the following question:
We know the following fact: Finite integral domain is a field.
What if we drop the assumption of finiteness? Consider the set $\mathbb{Z}_p[x]$ which is a collection of polynomials $a_0+a_1x+a_2x^2+\dots$ with coefficients from the field $\mathbb{Z}_p$. If we define usual addition and multiplication of polynomials it is easy to check that $\mathbb{Z}_p[x]$ is infinite integral domain. But it is not field since the element $x\in \mathbb{Z}_p[x]$ is non-zero and has not multiplicative inverse in $\mathbb{Z}_p[x]$.
Am I right?