Quotient 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. It is a specific example of a quotient, as viewed from the general setting of universal algebra. One starts with a ring "R" and a two-sided ideal "I" in "R", and constructs a new ring, the quotient ring , 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.
Formal quotient ring construction.
Given a ring formula_1 and a two-sided ideal formula_2 in formula_1, we may define an equivalence relation formula_4 on formula_1 as follows:
Using the ideal properties, it is not difficult to check that formula_4 is a congruence relation.
In case formula_6, we say that formula_11 and formula_12 are "congruent modulo" formula_2.
The equivalence class of the element formula_11 in formula_1 is given by
This equivalence class is also sometimes written as formula_17 and called the "residue class of formula_11 modulo formula_2".
The set of all such equivalence classes is denoted by ; it becomes a ring, the factor ring or quotient ring of formula_1 modulo formula_2, 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 formula_24 from formula_1 to defined by is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism.
Examples.
Alternative complex planes.
The quotients , , and are all isomorphic to R and gain little interest at first. But note that 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". This alternative complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent.
Furthermore, the ring quotient does split into and , so this ring is often viewed as the direct sum .
Nevertheless, an alternative complex number is suggested by j as a root of , compared to i as root of . This plane of split-complex numbers normalizes the direct sum by providing a basis 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.
Quaternions and alternatives.
Suppose "X" and "Y" are two, non-commuting, indeterminates and form the free algebra . Then Hamilton’s quaternions of 1843 can be cast as 
If is substituted for , 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 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 and constructing appropriate ideals.
Properties.
Clearly, if "R" is a commutative ring, then so is ; 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: "the ring homomorphisms defined on are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I". More precisely, given a two-sided ideal "I" in "R" and a ring homomorphism whose kernel contains "I", there exists precisely one ring homomorphism with (where "p" is the natural quotient map). The map "g" here is given by the well-defined rule for all "a" in "R". 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 induces a ring isomorphism between the quotient ring and the image im("f"). (See also: fundamental theorem on homomorphisms.)
The ideals of "R" and 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 (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 for the corresponding ideal in (i.e. ), the quotient rings and are naturally isomorphic via the (well-defined!) mapping .
The following facts prove useful in commutative algebra and algebraic geometry: for commutative, is a field if and only if "I" is a maximal ideal, while is an integral domain if and only if "I" is a prime ideal. A number of similar statements relate properties of the ideal "I" to properties of the quotient ring .
The Chinese remainder theorem states that, if the ideal "I" is the intersection (or equivalently, the product) of pairwise coprime ideals "I"1, ..., "Ik", then the quotient ring is isomorphic to the product of the quotient rings , . 
For algebras over a ring.
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.