What means $ℝ[i]$?

Question

From what i‘ve understood, a set, say $ℤ[i]$, denotes the set of all integers, in addition to i. ($i^2 = -1$) So for example, the complex number $1+i$ would be within that set.
In short, my question is, would then $ℝ[i]$ be the equivalent set as $ℂ$?

Answer 1

In general, $S=R[a]$ is a ring extension of $R$ by an element $a$ in a ring containing $R$, i.e., $S$ is the smallest ring which contains $R$ and $a$. Notice that all elements of $R[a]$ are necessarily in all rings containing $R$ and $a$, and it is itself a ring. Now apply this for $R=\Bbb{Z}$ or $R=\Bbb{R}$ in the ring $\Bbb{C}$ and for $a=i$. This answers your question "what means $\Bbb{R}[i]$?". Finally, $\Bbb{R}[i]\cong \Bbb{C}$ as rings (and as fields).