Quotient ring

Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.

[2][3] It is a specific example of a quotient, as viewed from the general setting of universal algebra.

(Quotient ring notation always uses a fraction slash "⁠

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.

as follows: Using the ideal properties, it is not difficult to check that

The set of all such equivalence classes is denoted by ⁠

⁠, if one defines (Here one has to check that these definitions are well-defined.

Compare coset and quotient group.)

is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism.

is called the dual number plane in geometric algebra.

It consists only of linear binomials as "remainders" after reducing an element of

This variation of a complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent.

⁠, so this ring is often viewed as the direct sum ⁠

This plane of split-complex numbers normalizes the direct sum

for 2-space where the identity of the algebra is at unit distance from the zero.

With this basis a unit hyperbola may be compared to the unit circle of the ordinary complex plane.

are two non-commuting indeterminates and form the free algebra ⁠

Substituting minus for plus in both the quadratic binomials also results in split-quaternions.

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminates

as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms.

are essentially the same as the ring homomorphisms defined on

⁠, there exists precisely one ring homomorphism

Indeed, this universal property can be used to define quotient rings and their natural quotient maps.

As a consequence of the above, one obtains the fundamental statement: every ring homomorphism

are closely related: the natural quotient map provides a bijection between the two-sided ideals of

are naturally isomorphic via the (well-defined) mapping ⁠

A number of similar statements relate properties of the ideal

The Chinese remainder theorem states that, if the ideal

is the intersection (or equivalently, the product) of pairwise coprime ideals ⁠

is isomorphic to the product of the quotient rings ⁠