Definition 1. The set of integers is
\[\Z = (\N \times \N) / {\sim},\]where $\sim$ is an equivalence relation on $\N \times \N$ defined by
\[(a, b) \sim (x, y) \enspace\lrimp\enspace a + y = b + x.\]Definition 2. Addition and multiplication on $\Z$ are defined by
\[\begin{align*} [(a, b)] + [(x, y)] &= [(a + x, b + y)], \\ [(a, b)] \cdot [(x, y)] &= [(ax + by,ay + bx)]. \end{align*}\]Definition 3. $\N$ is embedded into $\Z$ by
\[\N \to \Z, \quad n \mapsto [(n, 0)].\]Then we can say
\[\N \subseteq \Z.\]Definition 4. The following defines a well-order on $\Z$:
\[[(a, b)] \leq [(x, y)] \enspace\lrimp\enspace a + y \leq b + x.\]Definition 5. The set of non-zero integers is
\[\Z^* = \Z \setdif \{0\}.\]Definition 6. The set of positive integers is
\[\Z^+ = \{z \in \Z : z > 0\}.\]Note. $\Z^+ = \N^+$.
Definition 7. The set of negative integers is
\[\Z^- = \{z \in \Z : z < 0\}.\]Proposition 1. $(\Z, +, \cdot, 0, 1, \leq)$ is a well-ordered abelian ring.