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## Rings (Overview)

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.

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