Properties of Integers

Integers under Addition form Countably Infinite Abelian Group

The set of integers under addition $\left({\Z, +}\right)$ forms a countably infinite abelian group.

The set of integers is closed under addition:

$\forall a, b \in \Z: a + b \in \Z$

The operation of addition on the set of integers $\Z$ is associative:

$\forall x, y, z \in \Z: x + \left({y + z}\right) = \left({x + y}\right) + z$

The identity of integer addition is $0$:

$\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$

Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:

$\forall x \in \Z: \exists -x \in \Z: x + \left({-x}\right) = 0 = \left({-x}\right) + x$

The operation of addition on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x + y = y + x$

Integers are Countably Infinite

The set $\Z$ of integers is countably infinite.

Integers under Addition form Totally Ordered Group

Then the ordered structure $\left({\Z, +, \le}\right)$ is a totally ordered group.

Integers under Multiplication form Semigroup

The set of integers under multiplication $\left({\Z, \times}\right)$ is a semigroup.

Integer Multiplication is Closed

The set of integers is closed under multiplication:

$\forall a, b \in \Z: a \times b \in \Z$

Integer Multiplication is Associative

The operation of multiplication on the set of integers $\Z$ is associative:

$\forall x, y, z \in \Z: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

Integers under Multiplication form Monoid

The set of integers under multiplication $\left({\Z, \times}\right)$ is a monoid.

Integer Multiplication Identity is One

The identity of integer multiplication is $1$:

$\exists 1 \in \Z: \forall a \in \Z: a \times 1 = a = 1 \times a$

Integers under Multiplication form Countably Infinite Commutative Monoid

The set of integers under multiplication $\left({\Z, \times}\right)$ is a countably infinite commutative monoid.

Integer Multiplication is Commutative

The operation of multiplication on the set of integers $\Z$ is commutative:

$\forall x, y \in \Z: x \times y = y \times x$

Integers form Commutative Ring

The integers $\Z$ form a commutative ring under addition and multiplication.

Integers under Addition form Abelian Group

The set of integers under addition $\left({\Z, +}\right)$ forms an abelian group.

The operation of multiplication on the set of integers $\Z$ is distributive over addition:

$\forall x, y, z \in \Z: x \times \left({y + z}\right) = \left({x \times y}\right) + \left({x \times z}\right)$
$\forall x, y, z \in \Z: \left({y + z}\right) \times x = \left({y \times x}\right) + \left({z \times x}\right)$

Integer Multiplication has Zero

The set of integers under multiplication $\left({\Z, \times}\right)$ has a zero element, which is $0$.

Integers form Commutative Ring with Unity

The integers $\left({\Z, +, \times}\right)$ form a commutative ring with unity under addition and multiplication.

Integer Multiplication has Identity Element

The identity of integer multiplication is $1$:

$\exists 1 \in \Z: \forall a \in \Z: a \times 1 = a = 1 \times a$

Integers under Addition form Infinite Cyclic Group

The additive group of integers $\left({\Z, +}\right)$ is an infinite cyclic group which is generated by the element $1 \in \Z$.

Integers form Integral Domain

The integers $\Z$ form an integral domain under addition and multiplication.

Integers form Commutative Ring with Unity

The integers $\left({\Z, +, \times}\right)$ form a commutative ring with unity under addition and multiplication.

Ring of Integers has no Zero Divisors

The integers have no zero divisors:

$\forall x, y, \in \Z: x \times y = 0 \implies x = 0 \lor y = 0$

Integers form Totally Ordered Ring

The structure $\left({\Z, +, \times, \le}\right)$ is a totally ordered ring.

Integers form Ordered Integral Domain

The integers $\Z$ form an ordered integral domain under addition and multiplication.

Integers are Euclidean Domain

The integers $\Z$ with the mapping $\nu: \Z \to \Z$ defined as:

$\forall x \in \Z: \nu \left({x}\right) = \left \vert {x} \right \vert$

form a Euclidean domain.

Substructures and Superstructures

Additive Group of Integers is Subgroup of Rationals

Let $\left({\Z, +}\right)$ be the additive group of integers.

Let $\left({\Q, +}\right)$ be the additive group of rational numbers.

Then $\left({\Z, +}\right)$ is a normal subgroup of $\left({\Q, +}\right)$.

Additive Group of Integers is Subgroup of Reals

Let $\left({\Z, +}\right)$ be the additive group of integers.

Let $\left({\R, +}\right)$ be the additive group of real numbers.

Then $\left({\Z, +}\right)$ is a normal subgroup of $\left({\R, +}\right)$.

Additive Group of Integers is Subgroup of Complex

Let $\left({\Z, +}\right)$ be the additive group of integers.

Let $\left({\C, +}\right)$ be the additive group of complex numbers.

Then $\left({\Z, +}\right)$ is a normal subgroup of $\left({\C, +}\right)$.

Integers form Subdomain of Rationals

The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers.

Integers form Subdomain of Reals

The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of real numbers.