Congruence Rings in Modular Arithmetic

Revised Thinking

Definition: a≡b modm if m|(b-a), or, if b and a leave same remainder when divided by m. a,b,m integers.

c is called the modular sum of a & b if (a+b)≡c modm.

c is called the product of a & b if ab≡c modm.

For a fixed m=n, and ≡ replacing =:

The above defines a ring Zn for the integers 0,1,2,..n-1,

Or a ring of residue classes if the integers are divided into classes with same remainder 0,1,2,..n-1.

Ring: Addition, multiplication, associativity, commutativity, 0, 1, and distributivity.

Domain: Ring plus cancellation law (m prime).