Let d 
R[X].
The usual arithmetical rules imply the rules below for addition
and multiplication modulo d. First we specify two special elements:
- The element 0 + dR[X] is called the zero element of R[X]/(d) and
- the element 1 + dR[X] is called the unity or unit element.
We often simply denote these elements by 0 and 1, respectively.
Arithmetical rules with 0 and 1
For arbitrary a
R[X]/dR[X] we have
- a + 0 = 0 + a = a;
- a · 0 = 0 · a = 0;
- a · 1 = 1 · a = a;
- there exists (a unique)
b
R[X]/(d) with
a + b = b + a = 0 + (d).
The element b is the opposite of a and is written as -a.
Here are some more rules for arbitrary
a,
b, c
R[X]/(d).
General arithmetical rules
- a + b
= b + c
(commutativity of addition); - a ·
b = b
· a
(commutativity of multiplication); - (a + b)
+ c = a + (b + c)
(associativity of addition); - (a · b) · c =
a · (b · c)
(associativity of multiplication); - a · (b + c) =
a · b + a · c
(distributivity of multiplication over addition).