Module

A mathematical object in which things can be added together Commutatively by multiplying Coefficients and in which most of the rules of manipulating Vectors hold. A module is abstractly very similar to a Vector Space, although modules have Coefficients in much more general algebraic objects and use Rings as the Coefficients instead of Fields.

The additive submodule of the Integers is a set of quantities closed under Addition and Subtraction (although it is Sufficient to require closure under Subtraction). Numbers of the form $n\alpha\pm m\alpha$ for $n,m\in \Bbb{Z}$ form a module since,

$$n\alpha\pm m\alpha=(n\pm m)\alpha.$$

Given two Integers $a$ and $b$, the smallest module containing $a$ and $b$ is $\mathop{\rm GCD}\nolimits (a,b)$.

References

Foote, D. and Dummit, D. Abstract Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1990.