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Concept
Quotient ring
Summary
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 group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations. (Only the fraction slash "/" is used in quotient ring notation, not a horizontal fraction bar.)
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 and a two-sided ideal in , we may define an equivalence relation on as follows:
if and only if is in .
Using the ideal properties, it is not difficult to check that is a congruence relation.
In case , we say that and are congruent modulo .
The equivalence class of the element in is given by
This equivalence class is also sometimes written as and called the "residue class of modulo ".
The set of all such equivalence classes is denoted by ; it becomes a ring, the factor ring or quotient ring of modulo , if one defines
(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of is , and the multiplicative identity is .
The map from to defined by is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism.
The quotient ring R / {0} is naturally isomorphic to R, and R / R is the zero ring {0}, since, by our definition, for any r in R, we have that [r] = r + "R" := {r + b : b ∈ "R", which equals R itself. This fits with the rule of thumb that the larger the ideal I, the smaller the quotient ring R / I. If I is a proper ideal of R, i.e., I ≠ R, then R / I is not the zero ring.
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