Abstract Algebra/Rings, ideals, ring homomorphisms

Basic definitions


Definition 10.1:

A ring is a set   together with two binary operations   and   and two special elements, the unit   and the zero  , such that:

  1.   is an abelian group with respect to   with neutral element  .
  2.   is a monoid (that is, a group without inversion) with respect to   with neutral element  .
  3. The distributive laws hold:  ,  .

Examples 10.2:

  • The whole numbers   with respect to usual addition and multiplication are a ring.
  • Every field is a ring.
  • If   is a ring, then all polynomials over   form a ring. This example will be explained later in the section on polynomial rings.

Definition 10.3:

Let   be a ring. A left ideal of   is a subset   such that the following two things hold:

  1.   is a subgroup of  .
  2.  , where   (closedness by left multiplication).

Replacing closedness by left multiplication by closedness by right multiplication, we can define right ideals, and then both-sided ideals. If   is a both-sided ideal of  , we write  .

We'll now show an important property of the set of all ideals of a given ring, namely that it's inductive. This means:

Definition 10.4:

Let   be a partially ordered set (that is, the usual conditions transitivity, reflexivity and anti-symmetry are satisfied).   is called inductive if and only if every ascending chain of elements of   (that is, a sequence   in   such that  ) has an upper bound (that is, an element   such that  ).

With this definition, we observe:

Theorem 10.5:

If a commutative ring   is given, the set of all ideals   of  , partially ordered by inclusion (i.e.  , where we use the convention of Donald Knuth and denote the power set of a set   by  ) is inductive.




is an ascending chain of ideals, we set


and claim that  . Indeed, if  , find   such that   and  . Then set  , so that   since  . Similarly, if   and  , pick   such that  , whence   since  . 

Residue class rings


Definition and theorem 10.4:

Let   be a ring, and  . Then we define a relation   on   as follows:


This relation is an equivalence relation, and an equivalence class   shall be denoted by   for  . If we define an addition


and a multiplication


then these two are well-defined (i. e. independent of the choice of the representatives   and  ) and turn   into a ring, called the residue class ring with respect to the ideal  .


First, we check that   is an equivalence relation.

  1. Reflexiveness:   since   is an additive subgroup.
  2. Symmetry:   since inverses are in the subgroup.
  3. Transitivity: Let   and  . Then  , since a subgroup is closed under the group operation.

Then we check that addition and multiplication are well-defined. Let   and  . Then

  for certain  .



for these same  ; this is in   by closedness by left and right multiplication.

The ring axioms directly carry over from the old ring  . 

Ring homomorphisms


Definition 10.5:

Let   be rings. A ring homomorphism between the two is a map


such that:

  1. For all     and  .
  2.   (  is the unit of   and   of  ).