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Is Z a ring?

The set of integers is a ring. We will show how to prove that. There are many things which we need to check. First, we need to check if Z it is a group under addition.

Proof. (\mathbb{Z}, +) is a group: we have seen that here

\times is associative: so we need to check multiplication “\times” is associative. So let a,b,c \in \mathbb{Z}. Then (a \times b) \times c = a \times b \times c = a \times (b \times c).

Distributive laws hold in \mathbb{Z}: let a,b,c \in \mathbb{Z}. Then 
\begin{align*}
(a+b)\times c = a \times c + b \times c =  (a \times c) + (b \times c)
\end{align*}
and 
\begin{align*}
a \times (b + c) = a \times b + a \times c =  (a \times b) + (a \times c)
\end{align*}
Now we have shown that \mathbb{Z} is indeed a ring. It is easy to verify that Z is a commutative ring. The ring of integers also contains the identity, i.e. the element 1.
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