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!

$R[\alpha]$ and $R(\alpha)$ are different in general.

$R[\alpha]$ is the smallest subring of $S$ that contains both $R$ and $\alpha$.

$R(\alpha)$ is the smallest subfield of $S$ that contains both $R$ and $\alpha$. (*)

When $R$ is a field, $R[\alpha]=R(\alpha)$ iff $\alpha$ is algebraic over $R$.

For instance, $\mathbb Q[\sqrt2]=\mathbb Q(\sqrt2)$ but $\mathbb Q[\pi]\ne\mathbb Q(\pi)$. In fact, $\mathbb Q[\pi]\cong \mathbb Q[X]$ and $\mathbb Q(\pi)\cong \mathbb Q(X)$. So, $\mathbb Q[\pi]$ is not a field but $\mathbb Q(\pi)$ is.

(*) The notation $R(\alpha)$ is probably used only when $R$ is a field. One could argue that for instance $\mathbb Z(\sqrt2)$ should be equal to $\mathbb Q(\sqrt2)$, but I've never seen it done, and it's probably useless and confusing.