Properties of Integers
Contents
- 1 Additive Group of Integers is Countably Infinite Abelian Group
- 2 Integers under Addition form Totally Ordered Group
- 3 Integers under Multiplication form Semigroup
- 4 Integers under Multiplication form Monoid
- 5 Integers under Multiplication form Countably Infinite Commutative Monoid
- 6 Integers form Commutative Ring
- 7 Integers form Commutative Ring with Unity
- 8 Integers under Addition form Infinite Cyclic Group
- 9 Integers form Integral Domain
- 10 Integers form Totally Ordered Ring
- 11 Integers form Ordered Integral Domain
- 12 Integers are Euclidean Domain
- 13 Substructures and Superstructures
- 14 Also see
Additive Group of Integers is Countably Infinite Abelian Group
The set of integers under addition $\struct {\Z, +}$ forms a countably infinite abelian group.
Integer Addition is Closed
The set of integers is closed under addition:
- $\forall a, b \in \Z: a + b \in \Z$
Integer Addition is Associative
The operation of addition on the set of integers $\Z$ is associative:
- $\forall x, y, z \in \Z: x + \left({y + z}\right) = \left({x + y}\right) + z$
Integer Addition Identity is Zero
The identity of integer addition is $0$:
- $\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$
Inverses for Integer Addition
Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition:
- $\forall x \in \Z: \exists -x \in \Z: x + \left({-x}\right) = 0 = \left({-x}\right) + x$
Integer Addition is Commutative
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 $\struct {\Z, \times}$ 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:
- $\forall x, y, z \in \Z: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$
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 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:
- $\forall x, y, z \in \Z: x \times \left({y + z}\right) = \left({x \times y}\right) + \left({x \times z}\right)$
- $\forall x, y, z \in \Z: \left({y + z}\right) \times x = \left({y \times x}\right) + \left({z \times x}\right)$
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 $\struct {\Z, +}$ 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 $\left({\Z, +, \times}\right)$ 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 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 $\left({\Z, +, \times}\right)$ forms a subdomain of the field of real numbers.