Discussion of Problem Sets

Definition 3.1   A subset $A$ of $\mathbb{Z}$ is a module if it is closed under addition, subtraction, and contains zero.

Remark 3.2   Closure under addition technically follows from closure under subtraction, so is a redundant assumption in the above definition.

The following theorem follows from the fact that a subgroup of a cyclic group is necessarily cyclic. It is also the statement that $\mathbb{Z}$ is a PID.

Theorem 3.3   All modules are of the form $d\mathbb{Z}$.