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
