In this chapter, we will introduce an even more complex algebraic structure - the *ring*. In rings, not only one but two different binary operations are defined, usually referred to as "addition" and "multiplication". The following tabular overview indicates the properties of a ring in comparison to previous algebraic structures:

Algebra $(X,\ast)$ | Closure | Associativity | Neutral Element | Existence of Inverse | Cancellation | Commutativity | Distributivity. |
---|---|---|---|---|---|---|---|

Magma | ✔ | (✔) | (✔) | (✔) | (✔) | (✔) | n/a |

Semigroup | ✔ | ✔ | (✔) | (✔) | (✔) | (✔) | n/a |

Monoid | ✔ | ✔ | ✔ | (✔) | (✔) | (✔) | n/a |

Group | ✔ | ✔ | ✔ | ✔ | ✔ | (✔) | n/a |

Ring $(R,\oplus,\odot)$ | $\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ (✔) |
$\oplus$ ✔, $\odot$(✔) |
$\oplus$ ✔, $\odot$ (✔) |
$\oplus$ ✔, $\odot$ (✔) |
✔ |

Like it was the case for groups, in rings, the existence of inverse elements ensures the cancellation property. Moreover, the "addition" operation "$\oplus$" is always commutative, has a neutral element and inverse elements. Unfortunately, in general, these properties are not fulfilled for the "multiplication" operation "$\odot$". Rings, in which these properties also hold for multiplication are called *fields*, which we will introduce later.

Examples: 1

- Axiom: Axiom of Distributivity
- Definition: (Unit) Ring
- Definition: Commutative (Unit) Ring
- Definition: Subring
- Definition: Ring Homomorphism