Definition N28.1. A submodule of is called an ideal. That is, an ideal is a subset such that
is an abelian subgroup of ;
(multiplication by any takes to ).
Example N28.2. The ideals in are the (and in they are the ), i.e., each is generated by a single element. This is very special: and are “principal ideal domains”.
NOW BACK TO GROUPS (YAY!)
Classification of finitely generated abelian groups
Note. Think of these as modules, i.e., -modules, and use the methods of linear algebra over , since a -module is the same thing as an abelian group.
Theorem N28.3. Any finitely generated abelian group is isomorphic to a product of cyclic groups:
if (i.e., they are coprime)—this is rather useful.
Proposition N28.4 (Prop 1). If is a finitely generated -module, then there exist and with , i.e., there is an exact sequence