> [!NOTE] Definition (Modular operations) > Let $n\in \mathbb{Z}.$ We can define the following operations on the [[Congruence Class|congruence classes modulo n]]: > > Addition: $[a]_{n} +_{n} [b]_{n} = [a+b]_{n}$ > > Multiplication: $[a]_{n} \times [b]_{n} = [a \times b]_{n}$ > > Negation: $-_{n} [a_{n}] = [-a]_{n}$ > # Properties > [!NOTE] Theorem (Properties of modular operations) > We claim that they are [[Well-defined Function with Respect to Equivalence Relation|well defined]], [[Commutativity|commutative]], [[Associativity|associative]] and [[Distributivity|distributive]]. *Proof*. (Addition) First we show that addition is well-defined. Suppose $a'\equiv a \pmod{n}$ and $b'\equiv b \pmod{n}.$ Then $a'=a+xn$ and $b'=b+yn$ for some $x,y\in\mathbb{Z}.$ So $a'+b' =a'+b'+n(x+y)$ which shows that $a'+b' \equiv a+b \pmod{n}.$ Commutativity, associativity and distributivity follow from [[Integers|integer]] addition. Equivalently we say that $\mathbb{Z}/n\mathbb{Z}$ ([[Integers Modulo n]]) is a *commutative ring* wrt to modular addition and multiplication.