Definition#

Let \(n \geq 2\) be an integer. The set of integers modulo \(n\), denoted by \(\mathbb{Z}_n\), is the set of all equivalence classes of \(\mathbb{Z}\) with respect to the equivalance relation \(a \equiv b \: \text{(mod } n \text{)}\). We call these congruence classes modulo \(n\). Specifically, \(\mathbb{Z}_n = \{[0], [1], [2], ..., [n-1]\}\).

Example#

The elements of \(\mathbb{Z}_4\) are \([0]\), \([1]\), \([2]\), and \([3]\), where

\[\begin{aligned} \\ [0] &= \{..., -8, -4, 0, 4, 8, ...\} \\ [1] &= \{..., -7, -3, 1, 5, 9, ...\} \\ [2] &= \{..., -6, -2, 2, 6, 10, ...\} \\ [3] &= \{..., -5, -1, 3, 7, 11, ...\} \end{aligned}\]

Source#

Gregory T. Lee. Abstract Algebra. 1st ed. Springer Cham, 2018. pp. 28.