Field Extension Notation

Field Extension Notation

I've seen similar questions asked here, but I've not been able to find a
comprehensive answer.
I know that for a ring $R$, $R[X]$ denotes the ring of polynomials over
$R$ and $R(X)$ denotes the field of fractions of $R[X]$. But if $\alpha
\in S$, where $S \supseteq R$ are rings, what is the distinction between
$R[\alpha]$ and $R(\alpha)$? It is my understanding that if $R$ is a field
then $R[\alpha] \cong R(\alpha)$, but that this is not generally true for
any ring. Is this correct?
Adding to my confusion is the fact that $\mathbb Z[i]$ and $\mathbb Z(i)$
are used interchangeably, despite $\mathbb Z$ not being a field. However,
it's clear to me that they are equivalent ( i.e. $\mathbb Z[i] = \mathbb
Z(i) = \{ a + bi | a,b \in \mathbb Z \}$); is this because $i$ is
algebraic over $\mathbb Z$? Or for some other reason?
Thanks for your help!