# General Fundamentals

# Monoids and Groups

Definition

A monoid is a semi-group SS with has a unit element 11, such that 1s=s1=s1s = s1 = s for each sSs \in S.

Remark

If a semi-group SS lacks a unit element, we can adjoin a formal element 11 to produce a monoid S=S{1}S^{\prime} = S \cup \{1\} and asserting that 1s=s1=s1s = s1 = s for all ss in SS.

Theorem (Fundamental Theorem of Abelian Groups)

Every finitely-generated abelian group is isomorphic to a finite direct sum of cyclic subgroups.

Definition

An element of a group is torsion if it has finite order. THe set of torsion elements of an abelian group GG is clearly a subgroup, and we say that GG is torsion-free if its torsion subgroup is 00.

Since any cyclic group is isomorphic to either (Z,+)(\ZZ,+) or (Z/nZ,+)(\ZZ/n\ZZ,+) for some nn in Z\ZZ, we have:

::: corollary
Every finitely-generated abelian group is the direct sum of a torsion-free abelian group and a finite abelian group.
:::

# Algebras

Definition

Let CC denote a commutative ring. A CC-algebra (or algebra over CC) is a ring RR which is also a CC-module whose scalar multiplications satifies the extra prop


# References

[1] Rowen, L.H. Ring theory Volume 1; Pure and applied mathematics; Academic Press: Boston, 1988; ISBN 978-0-12-599841-3.