Definitions of the integers and their arithmetic: 1. A [[Set-theoretic construction of the integers|set theoretic construction]]. 2. [[Integer Axioms]]. # Properties **Algebraic properties**: $(\mathbb{Z},+)$ is an infinite [[Cyclic Group|cyclic group]] generated by $1$ or $-1.$ Also $(\mathbb{Z},+,\times)$ is a [[Commutative Ring|commutative ring]]. The [[Ideals of Integers|ideals of integers]] are the $n\mathbb{Z}$ for $n\in \mathbb{Z}.$ **Relations on $\mathbb{Z}$**: Take $a,b\in\mathbb{Z}$ we say that $a$ [[Divisibility in Integers|divides]] $b$ iff there exists $n\in\mathbb{Z}$ so that $b=an.$ Take $n\in \mathbb{Z},$ two integers are said to be [[Congruence Modulo n|congruent modulo n]] if their difference is divisible by $n.$