# General Fundamentals
# Monoids and Groups
A monoid is a semi-group with has a unit element , such that for each .
If a semi-group lacks a unit element, we can adjoin a formal element to produce a monoid and asserting that for all in .
Theorem (Fundamental Theorem of Abelian Groups)
Every finitely-generated abelian group is isomorphic to a finite direct sum of cyclic subgroups.
An element of a group is torsion if it has finite order. THe set of torsion elements of an abelian group is clearly a subgroup, and we say that is torsion-free if its torsion subgroup is .
Since any cyclic group is isomorphic to either or for some in , we have:
Every finitely-generated abelian group is the direct sum of a torsion-free abelian group and a finite abelian group.
Let denote a commutative ring. A -algebra (or algebra over ) is a ring which is also a -module whose scalar multiplications satifies the extra prop
 Rowen, L.H. Ring theory Volume 1; Pure and applied mathematics; Academic Press: Boston, 1988; ISBN 978-0-12-599841-3.