User:Ybab321/Sandbox/Notes

Algebraic structures I'm to know:

Semigroup

$\left({S, \circ}\right)$ where:

Monoid

A semigroup $\left({G, \circ}\right)$ where:

Group

A monoid $\left({G, \circ}\right)$ where:

Abelian Group

A group $\left({G, +}\right)$ where:

Ring

$\left({R, *, \circ}\right)$ where:

$\left({R, *}\right)$ is an Abelian group

$\left({R, \circ}\right)$ is a semigroup

Commutative ring

A ring $\left({R, *, \circ}\right)$ where:

Ring with unity

A ring $\left({R, *, \circ}\right)$ where:

$\left({R, \circ}\right)$ is a monoid

Division ring

A ring with unity $\left({R, *, \circ}\right)$ where:

$\left({R \setminus \left\{{0}\right\}, \circ}\right)$ is a group

Integral domain

A commutative ring with unity $\left({R, *, \circ}\right)$ where:

Field

A ring with unity $\left({R, *, \circ}\right)$ where:

$\left({R \setminus \left\{{0}\right\}, \circ}\right)$ is an Abelian group

 Vector space

$\left({G, +_G, \circ}\right)_K$ where:

$\left({G, +_G}\right)$ is an Abelian group

$\left({K, +_K, \times_K}\right)$ is a field

$\forall \lambda, \mu \in K: \forall \mathbf x, \mathbf y \in G:$