An _integer_ is a number belonging in the _[[Infinite set]] of integers_ $\{...,-2,-1,0,1,2...\},$ denoted by the symbol, $\mathbb{Z}.$^[The letter Z here comes from the German word _Zahlen_, which means, numbers.] ^056f6b # The integer Group The integer [group](Groups.md) is an [Infinite group](Infinite%20group.md) consisting of the [infinite set](Infinite%20set) of all integers and the $+$ operation. Thus we express this group as $\langle \mathbb{Z}, +\rangle.$ ^553333 We see that this is indeed a group since the [associative](Groups.md#^6afd7d) [binary operator](Groups.md#^15feff) is addition ($+$), with subtraction ($-$) as the [inverse](Groups.md#^34658d) of addition, and the number $0$ as the [identity element](Groups.md#^3c624f). The group of integers is also [closed](Groups.md#^7e5920) under the binary operation, since adding or subtracting any two integers gives another integer. ^f1a57f The integer group is also [Abelian](Abelian%20groups.md) since the $+$ operator commutes. The group $\langle \mathbb{Z}, +\rangle$ is a [subgroup](Subgroups.md) of [$\langle \mathbb{Q}, +\rangle,$](Rational%20numbers.md#The%20rational%20number%20group) where $\mathbb{Q}$ is the set of all rational numbers and we refer to the group as _the set of all rational numbers under addition_. ^5572e0 # The integer ring The _integer [rings,](Rings.md)_ like all rings, is equipped with an additional operator $\cdot$ and thus is defined as $\langle \mathbb{Z}, +,\cdot\rangle.$ In this case $\cdot$ is the multiplication operator. #MathematicalFoundations/Numbers #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Groups #MathematicalFoundations/Algebra/AbstractAlgebra/RingTheory/Rings