Definition:Modulo Multiplication/Definition 3

Definition
Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:
 * $\Z_m = \set {0, 1, \ldots, m - 1}$

The operation of multiplication modulo $m$ is defined on $\Z_m$ as:
 * $x \cdot_m y := x y - k m$

where $k$ is the largest integer such that $k m \le x y$.

Also denoted as
Although the operation of multiplication modulo $m$ is denoted by the symbol $\times_m$, if there is no danger of confusion, the conventional multiplication symbols $\times, \cdot$ etc. are often used instead.

Also see

 * Equivalence of Definitions of Modulo Multiplication