# Quotient ring

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 groups of group theory and the quotient spaces of linear algebra.^{[2]}^{[3]} 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.

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.

The quotients **R**[*X*] / (*X*), **R**[X] / (*X* + 1), and **R**[*X*] / (*X* − 1) are all isomorphic to **R** and gain little interest at first. But note that **R**[*X*] / (*X*^{2}) is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of **R**[*X*] by *X*^{2}. This alternative complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent.

Suppose *X* and *Y* are two, non-commuting, indeterminates and form the free algebra **R**⟨*X*, *Y*⟩. Then Hamilton’s quaternions of 1843 can be cast as

If *Y*^{2} − 1 is substituted for *Y*^{2} + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in *both* the quadratic binomials also results in split-quaternions. The anti-commutative property *YX* = −*XY* implies that *XY* has as its square

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminates **R**⟨*X*, *Y*, *Z*⟩ and constructing appropriate ideals.

Clearly, if *R* is a commutative ring, then so is *R* / *I*; the converse, however, is not true in general.

The natural quotient map *p* has *I* 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.

The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: . More precisely, given a two-sided ideal *I* in *R* and a ring homomorphism *f* : *R* → *S* whose kernel contains *I*, there exists precisely one ring homomorphism *g* : *R* / *I* → *S* with *gp* = *f* (where *p* is the natural quotient map). The map *g* here is given by the well-defined rule *g*([*a*]) = *f*(*a*) for all *a* in *R*. Indeed, this universal property can be used to *define* quotient rings and their natural quotient maps.

*the ring homomorphisms defined on R / I are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I*

As a consequence of the above, one obtains the fundamental statement: every ring homomorphism *f* : *R* → *S* induces a ring isomorphism between the quotient ring *R* / ker(*f*) and the image im(*f*). (See also: fundamental theorem on homomorphisms.)

The ideals of *R* and *R* / *I* are closely related: the natural quotient map provides a bijection between the two-sided ideals of *R* that contain *I* and the two-sided ideals of *R* / *I* (the same is true for left and for right ideals). This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if *M* is a two-sided ideal in *R* that contains *I*, and we write *M* / *I* for the corresponding ideal in *R* / *I* (i.e. *M* / *I* = *p*(*M*)), the quotient rings *R* / *M* and (*R* / *I*) / (*M* / *I*) are naturally isomorphic via the (well-defined!) mapping *a* + *M* ↦ (*a* + *I*) + *M* / *I*.

The Chinese remainder theorem states that, if the ideal *I* is the intersection (or equivalently, the product) of pairwise coprime ideals *I*_{1}, ..., *I _{k}*, then the quotient ring

*R*/

*I*is isomorphic to the product of the quotient rings

*R*/

*I*,

_{n}*n*= 1, ...,

*k*.

An associative algebra *A* over a commutative ring *R* is a ring itself. If *I* is an ideal in *A* (closed under *R*-multiplication), then *A* / *I* inherits the structure of an algebra over *R* and is the **quotient algebra**.