We define the following equivalence relation on $\N\times\N$:
\[(a,b)\sim(x,y) \enspace\lrimp\enspace a+y=b+x.\]The set of integers is defined as
\[\Z = (\N\times\N)/{\sim}.\]Addition and multiplication on $\Z$ are defined by
\[[(a,b)]+[(x,y)] = [(a+x,b+y)]\]and
\[[(a,b)]\cdot[(x,y)] = [(ax+by,ay+bx)].\]$\N$ is embedded into $\Z$: We write
\[0 := [(0,0)]\]and for every $n\in\N^+$,
\[n := [(n,0)]; \quad -n := [(0,n)].\]Then we can say
\[\N\subseteq\Z.\]The following defines a well-order on $\Z$:
\[[(a,b)]\leq[(x,y)] \enspace\lrimp\enspace a+y\leq b+x.\]The set of non-zero integers is
\[\Z^* = \Z\setminus\{0\}.\]The set of negative integers is
\[\Z^- = \{z\in\Z : z<0\}.\]
If $a$, $b\in\Z$, we write
\[\llbra a,b\rrbra = \{k\in\Z:a\leq k\leq b\}.\]
If $a\leq b$,
\[\llbra a,b\rrbra = \{a,a+1,\cdots,b\}.\]If $b<a$,
\[\llbra a,b\rrbra = \varnothing.\]If $n\in\Z$, we write
\[\llbra n\rrbra = \llbra 1,n\rrbra.\]
If $n\leq 0$,
\[\llbra n\rrbra = \varnothing.\]
$(\Z,+,\cdot,0,1)$ is an abelian ring.