TITLE: Is the following a ring?
QUESTION [5 upvotes]: Is $\{a+bi: a,b \in \mathbb{Z} \} $ and $i^2=-1$, a ring under 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.

REPLY [6 votes]: Yes it is. This ring is known as the Gaussian integers, and you can learn a great deal about it on Wikipedia.

REPLY [4 votes]: The multiplicative identity is $1+0i$, the additive identity is $0+0i.$
Associativity (and commutativity) of addition and multiplication follow from the same properties of complex numbers in general (which you should verify if you haven't done so before!), as does distributivity.
You can easily check that it is multiplicatively closed by expanding $(a+bi)(c+di)$ for $a,b,c,d\in\mathbb Z.$
The additive inverse of $a+bi$ is $-a-bi.$