quotient ring

Definition

Let be a ring and be an ideal. Let be the quotient of the abelian group by the (normal) subgroup (since is a subgroup under addition). This makes an abelian group with elements

such that

Then we can take the product

which gives a ring structure. We call the quotient of by .

Quotient map

This product defined above is well-defined and gives the structure of a ring with identity . The canonical map

is a surjective homomorphism with kernel .

Relations to ideals and subrings

• Every ideal is the kernel of a ring homomorphism.

• Every subring is the image of a ring homomorphism (this is trivial).

Universal property

The universal property of the quotient of rings is a special case of the map above being a categorical quotient.

Let be rings and and ideal of . For every ring homomorphism with , there exists a unique ring homomorphism which makes the following diagram commute:

Proof

Given such an , define .

This is well-defined because if , that is , then

Clearly . Moreover, is a ring homomorphism:

and