what does this notation mean. Rings

Question

As I have started on learning rings, I saw the notation $\mathbb{Z}[i]$ was used in the context of '$\mathbb{Z}[i]$ is a commutative ring'. What does this notation mean? Is it complex numbers where the coefficients are integer values?

Answer 1

Yes, $\mathbb Z[i]=\{a+bi\mid a,b\in\mathbb Z\}$. More generally, if $\omega\in\mathbb C$, $\mathbb{Z}[\omega]$ is the smallest subring of $\mathbb C$to which $\omega$ belongs.

Answer 2

It denotes the set of Gaussian integers, i.e. the set of elements in $\mathbf C$ of the form $a+bi$, where $a,b\in \mathbf Z$.

It also can be characterised as the ring of integers of the field extension $\mathbf Q(i)/\mathbf Q$. It is not only a commutative ring, it is a Euclidean domain.