Residue class ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra. One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I, whose elements are the cosets of I in R subject to special + and ⋅ operations.

Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well as from the more general 'rings of quotients' obtained by localization.

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by

This equivalence class is also sometimes written as a mod I and called the "residue class of a modulo I".

The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of R modulo I, if one defines

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of R/I is (0 + I) = I, and the multiplicative identity is (1 + I).

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