1 Set With One Operation

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Studiengang Kommunikationsinformatik (Master) Prof. Dr.–Ing. Damian Weber

Handout – Definitions

LetG be a set with|G| ≥1.

A binary operation ◦on Gis a map

◦:G×G−→G.

(G,◦) is called a semigroup if◦is associative:

(a◦b)◦c=a◦(b◦c) ∀a, b, c∈G.

A semigroup (G,◦) is called a monoid if there exists a neutral element ewith a◦e=e◦a=a ∀a∈G.

A monoid (G,◦) is called a group if, for each a∈ G, there exists an inverse element a⁻¹ with

a⁻¹◦a=e.

A group (G,◦) is called an abelian grouporcommutative group if a◦b=b◦a ∀a, b∈G.

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LetR be a set with |R| ≥1.

Let ,,+” be an operation which makes (R,+) an abelian group. The neutral element ofR is called 0.

Let ,,·” be an operation which makes (R,·) an semigroup.

Then R is called aringifa(b+c) =ab+acand (b+c)a=ba+cafor alla, b, c∈R.

The ringR is called commutative ifab=ba for alla, b∈R.

The ringR is called ring with identityif 1∈R witha·1 = 1·a=afor all a∈R.

LetF be a set with {0,1} ∈F and 06= 1.

LetF be a commutative ring with identity.

Let (F \ {0},·) be an abelian group.

Then F is called afield.

The smallest numberm∈Nsuch that

1 + 1 + 1 +. . .+ 1

| {z }

m

= 0

is called the characteristic of the field.

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