Section 7.6
Residue class rings
In this section, arithmetic modulo an integer n or modulo a polynomial d is generalized to arithmetic modulo an ideal.
Let I be an ideal in the commutative ring R.
Two elements a, b
R are called
congruent modulo I if their difference a - b
belongs to I.
Proposition
Congruence modulo I is an equivalence relation.
Congruence modulo I of a and b is often written as a = b mod I. An equivalence class is called a residue class. The set of all residue classes is denoted by R/I. An element of R/I is denoted by a + I when we are precise, and simply by a if there is no danger of confusion.
Theorem
The set R/I inherits from R the following ring structure:
- addition: (a + I) + (b + I) = (a + b) + I,
- multiplication: (a + I) * (b + I) = (a * b) + I,
- unit element: 1 + I,
- zero element: 0 + I.
The ring R/I is called the residue class ring or quotient ring of R modulo I.