Addition modulo operation

Description:

• $a{+}_{m}b=(a+b)\mathrm{mod}m$
• denoted by ${+}_m$
• The operations ${+}_m$ satisfies many of the same properties of ordinary addition of integers.
  1. Closure: If $a$ and $b$ belong to ${\mathbb{Z}}_m$, then $a{+}_{m}b$ belongs to $\mathbb{Z}m$.
  2. Associativity: If $a,b,$ and $c$ belong to $\mathbb{Z}m$, then $(a{+}_{m}b){+}_{m}c=a{+}_m(b{+}_{m}c)$
  3. Commutativity: If $a$ and $b$ belong to $\mathbb{Z}m$, then $a{+}_{m}b=b{+}_{m}a$
  4. Identity element: The element 0 is identity element for addition modulo m. That is, if a belongs to $\mathbb{Z}m$, then $a{+}_{m}0=0{+}_{m}a=a$.
  5. Additive inverses: If $a\ne 0$ belongs to $\mathbb{Z}m$, then $m-a$ is an additive inverse of a modulo m and 0 is its own additive inverse. That is, $a{+}_m(m-a)=0$ and $0{+}_{m}0=0$
• ${\mathbb{Z}}_m$ with modular addition is said to be a Commutative group