## Quotient Rings (Pt. I)

**Point of Post: **In this post we give the construction of quotient rings by ideals as well as the canonical projection .

*Motivation*

Intuitively what we’d like to do is make a construction similar to that of a quotient group but for rings. Namely, for some subset of a ring we’d like to define a ring structure on the set of equivalence classes if and only if , which ‘collapses’ down to a point in such a way that the canonical map is a ring morphism. We know from group theory that since is a group morphism we must have that , but since is abelian this is equivalent to . So, the obvious question then is what multiplicative conditions must be imposed on ? What we shall see is that must be an ideal of for this construction to work. A little experimentation actually makes this obvious since and so and thus for any and we have and so and similarly .

*Quotient Ring*

Let be a ring and . Since and is abelian (so that ) we may form the quotient group . Recall that the elements of are the sets written and their group structure is defined by . We know from basic group theory that since is abelian so is .

What we now claim is that the ‘multiplication’ is well-defined on . Indeed, if and then for and for . We see then that

the last conclusion made since (since is an ideal and ) and so and so the last claim follows by definition. What we now claim is that with this multiplication and addition becomes a ring. Indeed:

**Theorem: ***Let be a ring and , then with the addition and multiplication is a ring.*

**Proof: **We already know from group theory that is a group and we know from the above discussion that the multiplication is a binary operation, so it suffices to show that it’s associative and left and right distributive. The first of these is clear since

To see left distributivity we note that

and right distributivity is done similarly.

With this structure we call the ring *a quotient ring* or when we want to be more precise *the quotient ring of by *. We note that this structure (by construction) satisfies our desired result that:

**Theorem: ***Let be a ring and then the map is an epimorphism with .*

This map is called the *canonical projection of onto *or *canonical projection *for short. From the existence of the canonical projection we are able to conclude that inherits many of the properties of . For example if is commutative or unital then so is since these are both properties preserved under epimorphism. But, we are also able to gleam from this theorem the converse of a previously noted theorem. In particular, we can now legitimately say:

**Theorem: ***Let be a ring, then is an ideal if and only if there exists a ring and morphism for which .*

We note next that there is a certain ‘mapping property’ that the canonical projection possess, namely:

**Theorem: ***Let and be rings, and a ring morphism with . Then, there exists a unique morphism for which .*

**Proof: **Consider the mapping defined by . This is well-defined since if then for and so . This is a morphism since

and evidently for all and so .

and

Since uniqueness is clear the conclusion follows.

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

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