(related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring $\mathbb Z_m$)
Let $m > 0$ be a positive integer and let $a_1,\ldots,a_r$ be integers. Then the sum and the product of the congruences $a_i(m)$, $i=1,\ldots,r$ are defined by
$$\sum_{i=1}^ra_i(m):=\left(\sum_{i=1}^ra_i\right)(m),$$
$$\prod_{i=1}^ra_i(m):=\left(\prod_{i=1}^ra_i\right)(m).$$
For any integer $a$, we set the power of the congruence $a(m)$ by
$$(a(m))^r:=(a^r)(m).$$
Proofs: 1
