Multiplication modulo

Description:

• $a{.}_{m}b=(a\cdot b)\mathrm{mod}m$
  • denoted by ${.}_m$
• The operations ${.}_m$ satisfies many of the same properties of ordinary multiplication 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 $a{.}_{m}1=1{.}_{m}a=a$.
  5. Distributivity: If $a,b,$ and $c$ belong to ${\mathbb{Z}}_m$, then $a{.}_m(b{+}_{m}c)=(a{.}_{m}b){+}_m(a{.}_{m}c)$ and $(a{+}_{m}b){.}_{m}c=(a{.}_{m}c){+}_m(b{.}_{m}c)$.
• ${\mathbb{Z}}_m$ with modular addition and multiplication is said to be a Commutative ring
• The multiplicative inverses has not been included ${\mathbb{Z}}_m$ as it may not be exists
  • (E.g., no multiplicative inverse of 2 modulo 6).
• For some certain value of $m$, the multiplicative inverses exists (for all element not 0), in this case, ${\mathbb{Z}}_m$ is a field.