Group

Group is a Set# of elements \(S\), together with one operation \(+\), which has the following properties:

Closure under addition, which means if \(a\) and \(b\) belong to \(S\), then \(a + b\) is also in \(S\).
Associativity of addition, which means \(\forall a, b, c \in S, a + (b + c) = (a + b) + c\).
Additive identity, which means \(\exist x \in R, \forall a \in S, a + x = x + a = a\).
Additive inverse, which means \(\forall a \in S, \exist -a \in S, a + (-a) = (-a) + a = 0\).

If the group is also commutative, then it is also an Abelian Group#.

If every element is a power of some fixed element, such as \(\exist a \in S, \forall b \in S, b = a^k\), then the group is also a cyclic group. In this cause, \(a\) is also a generator of the group.

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Ring

Ring is a Set# \(R\), together with two operations \(+\) and \(*\), which has the following properties including those in #Group and #Abelian Group:
Modular Arithmetic

Note: The #inverse of the \(a\) could be found in the table by finding the matched row or column with value 0. For example, in the column \(4\), the value 0 appear in the row \(4\), thus \(4\) is the inverse of \(4\) in modulo 8.
Integral Domain

Integral Domain is a #Ring which satisfies the following properties including those in #Group, #Abelian Group and #Communicative Ring.
Field

Field is a Set# \(S\), together with two operations \(+\) and \(*\), which has the following properties including those in #Group, #Abelian Group, #Ring, #Communicative Ring, and #Integral Domain:
Communicative Ring

Communicative Ring is a #Ring where its multiplicative operation is communicative, as shown below. It also satisfies the properties from #Group and #Abelian Group.
Algebraic Structure

Group#
Abelian Group

Abelian Group is a #Group which also satisfies the following property:

#math