# # General Fundamentals

## # Monoids and Groups

Definition

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

Remark

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

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 $G$ is clearly a subgroup, and we say that $G$ is torsion-free if its torsion subgroup is $0$.

Since any cyclic group is isomorphic to either $(\ZZ,+)$ or $(\ZZ/n\ZZ,+)$ for some $n$ in $\ZZ$, 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 $C$ denote a commutative ring. A $C$-algebra (or algebra over $C$) is a ring $R$ which is also a $C$-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.

Last Updated: 12/21/2019, 3:04:48 AM