Is $\displaystyle \{a+bi:a,b \in \mathbb{Z}\}$ where $\displaystyle i^2=-1$ a ring with the usual operations of addition and multiplication?

It needs to be closed under addition and multiplication which it seems like

Addition needs to be commutative and associative which is quite straightforward.

It must have an identity element which I am not sure about

and it must have an additive inverse which I am also unclear about

Multiplication must be associative which it should be and then it must be distributive over addition meaning that a*(b+c)=a*b+a*c which i also think it is.