Is Z a ring?

The Mathematician — Tue, 04 Oct 2022 13:00:20 +0000

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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Is Z a ring?