## Algebraic Structures

If on a set one or more binary operations are defined, it is called an **algebraic structure** (or shortly speaking an **algebra**). Usually, algebraic structures are denoted by the symbol of the set in conjunction with all its maps under consideration. For instance, if \(S\) is the set of natural numbers \(\mathbb N\), integers \(\mathbb Z\), real numbers \(\mathbb R\), complex numbers \(\mathbb C\), congruence classes with the modulus \(n\) \(\mathbb Z_{n\mathbb Z}\), etc. then the maps

\[+ :=\begin{cases}S \times S & \mapsto S \\(x,y) & \mapsto x + y\\\end{cases}\]

and

\[\cdot :=\begin{cases}S \times S & \mapsto S \\(x,y) & \mapsto x \cdot y\\\end{cases}\]

define the respective algebraic structures \((\mathbb N,+ ,\cdot)\), \((\mathbb Z,+ ,\cdot)\), \((\mathbb R, +,\cdot)\), \((\mathbb C, +,\cdot)\), \((\mathbb Z_{n\mathbb Z}, +,\cdot),\), etc.

| | | | Contributors: *bookofproofs*

## 1.Groupoids

## 2.Semigroups

## 3.Groups

## 4.Semi-Rings

## 5.Rings

## 6.Polynomial Rings and Irreducibility

## 7.Fields

## 8.Vector Spaces

## 9.**Definition**: Binary Operation

## 10.**Definition**: Module

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