# Properties of Integers

## Additive Group of Integers is Countably Infinite Abelian Group

The set of integers under addition $\struct {\Z, +}$ 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:

$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$

### Integer Addition Identity is Zero

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:

$\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$

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 $\struct {\Z, +, \le}$ is a totally ordered group.

## Integers under Multiplication form Semigroup

The set of integers under multiplication $\struct {\Z, \times}$ 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:

$\forall x, y, z \in \Z: x \times \paren {y \times z} = \paren {x \times y} \times z$

## Integers under Multiplication form Monoid

The set of integers under multiplication $\struct {\Z, \times}$ 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 $\struct {\Z, \times}$ 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 set of integers $\Z$ forms a commutative ring under addition and multiplication.

### Integers under Addition form Abelian Group

The set of integers under addition $\struct {\Z, +}$ forms an abelian group.

### Integer Multiplication Distributes over Addition

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

$\forall x, y, z \in \Z: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$
$\forall x, y, z \in \Z: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$

### Integer Multiplication has Zero

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

## Integers form Commutative Ring with Unity

The integers $\struct {\Z, +, \times}$ 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 $\struct {\Z, +}$ 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 $\struct {\Z, +, \times}$ 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 $\struct {\Z, +, \times, \le}$ 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:

$\forall x \in \Z: \map \nu x = \size x$

form a Euclidean domain.

## Substructures and Superstructures

### Additive Group of Integers is Normal Subgroup of Rationals

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\Q, +}$ be the additive group of rational numbers.

Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.

### Additive Group of Integers is Normal Subgroup of Reals

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\R, +}$ be the additive group of real numbers.

Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.

### Additive Group of Integers is Normal Subgroup of Complex

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\C, +}$ be the additive group of complex numbers.

Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\C, +}$.

### 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 $\struct {\Z, +, \times}$ forms a subdomain of the field of real numbers.