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Factor ring

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings.

Given a ring   and a two-sided ideal , define the following equivalence relation ~ on :  

(1)

The equivalence is a congruence relation. If , then~ we say that and are congruent modulo . The equivalence class of the element is given by

(2)

This equivalence class is also written as or simply and it called the residue class of modulo .

The set of all such equivalence classes is denoted by , and it  becomes a ring, the so-called the factor ring or quotient ring or residue class ring of R modulo I, if we define its operation as follows

(3)

(4)

These definitions are well-defined and  the corresponding structure on is a ring. The ring is a ring with (multiplicative) identity, namely . The zero-element of is .

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

The natural quotient map provides a bijection between the two-sided ideals of R that contain a and the two-sided ideals of .

Theorem. There exists a one-to-one correspondence (bijection) preserving the inclusion between the ideals that contain and ideals of the factor ring .

The same assertion is also true for left and for right ideals containing .

The natural quotient map has the so-called universal property in the category of homomorphisms whose kernel contains the ideal . This means:

Theorem. Let be a ring homomorphism. Its kernel is an ideal of  . If the kernel contains the ideal , then there exists precisely one ring homomorphism such that , where is the natural quotient map.

The map is given by the rule for all .

Previous theorem can be also formulated in the following way:

Theorem: If is a ring-homomorphism whose kernel contains , and is the natural quotient map, then there exists a unique ring-homomorphism making the following diagram commutative

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