exponent of module

Definition

Let be an R-module. An element is called an exponent of (respectively ) if

Importance

Exponents play an important role in classifying finitely generated torsion modules over a PID. (see PID modules for more).

For prime, the submodule of elements of such that their exponents are of the form is denoted . This notation comes from abelian groups (i.e. -modules) where would be called an abelian p-group.