group theory definition

properties

Before introduce group definition, let me list those properties mentioned.

1. Closure:
   For all $a, b \in G$, the relation $a \bullet b \in G$ holds.
2. associative:
   For all $a, b, c \in G$, the equation $(a \bullet b) \bullet c = a \bullet (b \bullet c)$ holds.
3. Commutative
   For all $a, b \in G$, $a \bullet b = b \bullet a$.
4. Identity element:
   There exists an element $e \in G$, such that for all elements $a \in G$, the equation $e \bullet a = a$ holds.
5. Inversible:
   For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a \bullet a^{-1} = e$, where $e$ is the identity element.
6. Distributive:
   For all $a, b, c \in G$, the equation $(a+b) \bullet c=a \bullet c+b \bullet c$ holds.

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group(G,•)

consists of a set of elements together with an operation $\bullet$ such that:

property		$\bullet$
clojure		$\checkmark$
associative		$\checkmark$
identity		$\checkmark$
inversible		$\checkmark$
commutative		$\times$
distributive		$\times$

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abelian group (A, •)

Abelian group is a group, but also have one additional property:

property		$\bullet$
clojure		$\checkmark$
associative		$\checkmark$
identity		$\checkmark$
inversible		$\checkmark$
commutative		$\checkmark$
distributive		$\times$

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rings

A commutative ring with unity $(R,+,*)$ is an algebraic structure consisting of a set of elements R together with two binary operations denoted + and * which satisfy the follow properties for all elements in R:

property	$+$	$*$
clojure	$\checkmark$	$\checkmark$
associative	$\checkmark$	$\checkmark$
commutative	$\checkmark$	$\checkmark$
identity	$\checkmark$	$\checkmark$
inversible	$\checkmark$	N/A
distributive	$\times$	$\checkmark$

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ideal ring

Let $(R,+,*)$ be a ring; A non-empty subset $I$ of $R$ called a ideal of the ring if:

1. $(I,+)$ is a group.
2. $i*r \in I$ for all $i \in I$ and $r \in R$.

property		$+$
clojure		$\checkmark$
associative		$\checkmark$
identity		$\checkmark$
inversible		$\checkmark$
commutative		$\times$
distributive		$\times$

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field

field is a set of elements which is closed under two binary operations, which we denote by $+$ and $\times$.

property	$+$	$\times$
clojure	$\checkmark$	$\checkmark$
associative	$\checkmark$	$\checkmark$
commutative	$\checkmark$	$\checkmark$
identity	0	1
inversible	$\checkmark$	!0
distributive	$\times$	$\checkmark$