# Category:Integers

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This category contains results about Integers.

Definitions specific to this category can be found in Definitions/Integers.

The numbers $\set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ are called the **integers**.

This set is usually denoted $\Z$.

An individual element of $\Z$ is called **an integer**.

## Subcategories

This category has the following 19 subcategories, out of 19 total.

### A

### C

### D

### E

### I

### M

### N

### O

### R

### S

## Pages in category "Integers"

The following 129 pages are in this category, out of 129 total.

### A

### C

### D

### E

### H

### I

- Idempotent Elements for Integer Multiplication
- Inclusion of Natural Numbers in Integers is Epimorphism
- Infimum of Set of Integers equals Smallest Element
- Infimum of Set of Integers is Integer
- Integer Addition Identity is Zero
- Integer Addition is Associative
- Integer Addition is Cancellable
- Integer Addition is Closed
- Integer Addition is Commutative
- Integer Divisor is Equivalent to Subset of Ideal
- Integer Divisor Results
- Integer Multiples Closed under Addition
- Integer Multiples Closed under Multiplication
- Integer Multiples form Commutative Ring
- Integer Multiples Greater than Positive Integer Closed under Addition
- Integer Multiples Greater than Positive Integer Closed under Multiplication
- Integer Multiplication Distributes over Addition
- Integer Multiplication Distributes over Addition/Corollary
- Integer Multiplication Distributes over Subtraction
- Integer Multiplication has Zero
- Integer Multiplication Identity is One
- Integer Multiplication is Associative
- Integer Multiplication is Closed
- Integer Multiplication is Commutative
- Integer Multiplication is Well-Defined
- Integer Subtraction is Closed
- Integers are Countably Infinite
- Integers are Euclidean Domain
- Integers are not Close Packed
- Integers do not form Field
- Integers form Commutative Ring
- Integers form Commutative Ring with Unity
- Integers form Integral Domain
- Integers form Ordered Integral Domain
- Integers form Subdomain of Rationals
- Integers form Subdomain of Reals
- Integers form Subring of Reals
- Integers form Totally Ordered Ring
- Integers form Unique Factorization Domain
- Integers under Addition form Abelian Group
- Integers under Addition form Infinite Cyclic Group
- Integers under Addition form Monoid
- Integers under Addition form Semigroup
- Integers under Addition form Totally Ordered Group
- Integers under Multiplication do not form Group
- Integers under Multiplication form Countably Infinite Commutative Monoid
- Integers under Multiplication form Countably Infinite Semigroup
- Integers under Multiplication form Monoid
- Integers under Multiplication form Semigroup
- Integers under Subtraction do not form Group
- Intersection of Integer Ideals is Lowest Common Multiple
- Inverses for Integer Addition
- Invertible Integers under Multiplication
- Invertible Integers under Multiplication/Corollary 1
- Irreducible Elements of Ring of Integers

### N

- Natural Numbers are Non-Negative Integers
- Natural Numbers form Subsemiring of Integers
- Natural Numbers Set Equivalent to Ideals of Integers
- Natural Numbers under Multiplication form Subsemigroup of Integers
- Non-Zero Integer has Unique Positive Integer Associate
- Non-Zero Integers are Cancellable for Multiplication
- Non-Zero Integers Closed under Multiplication
- Non-Zero Integers under Multiplication are not Subgroup of Reals
- Nth Root of Integer is Integer or Irrational

### P

- Parity of Integer equals Parity of Positive Power
- Pointwise Addition on Integer-Valued Functions is Associative
- Pointwise Addition on Integer-Valued Functions is Commutative
- Pointwise Multiplication on Integer-Valued Functions is Associative
- Pointwise Multiplication on Integer-Valued Functions is Commutative
- Polynomials in Integers is not Principal Ideal Domain
- Polynomials in Integers is Unique Factorization Domain
- Polynomials in Integers with Even Constant Term forms Ideal
- Positive Integer is Well-Defined
- Prime Ideals of Ring of Integers
- Properties of Integers

### Q

### R

### S

- Set of Even Integers is Equivalent to Set of Integers
- Set of Integers Bounded Above by Integer has Greatest Element
- Set of Integers Bounded Above has Greatest Element
- Set of Integers Bounded Below by Integer has Smallest Element
- Set of Integers Bounded Below has Smallest Element
- Set of Integers can be Well-Ordered
- Set of Integers is not Well-Ordered by Usual Ordering
- Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition
- Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers
- Strictly Increasing Infinite Sequence of Positive Integers is Cofinal in Natural Numbers
- Subgroup of Integers is Ideal
- Subring of Integers is Ideal
- Subrings of Integers are Sets of Integer Multiples
- Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup/Examples/Non-Zero Integers in Non-Zero Reals under Multiplication
- Subtraction on Integers is Extension of Natural Numbers
- Sum of Integer Ideals is Greatest Common Divisor
- Supremum of Set of Integers equals Greatest Element
- Supremum of Set of Integers is Integer