# Category:Integers

Jump to navigation
Jump to search

This category contains results about **Integers** in the context of **abstract algebra**.

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 $\Z$.

## Subcategories

This category has the following 29 subcategories, out of 29 total.

### A

### C

### D

### E

### I

- Integer Subtraction (3 P)
- Integers do not form Field (3 P)
- Integral Ideals (4 P)

### L

### M

### N

### O

- Orderings on Integers (12 P)

### S

- Square-Free Integers (7 P)

## Pages in category "Integers"

The following 89 pages are in this category, out of 89 total.

### C

### D

### E

### H

### I

- Inclusion of Natural Numbers in Integers is Epimorphism
- Infimum of Set of Integers equals Smallest Element
- Infimum of Set of Integers is Integer
- Integer Divisor is Equivalent to Subset of Ideal
- Integer Divisor Results
- Integer Multiples form Commutative Ring
- Integers are Countably Infinite
- Integers are Euclidean Domain
- Integers are not Densely Ordered
- 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 Subtraction form Magma
- Intersection of Integer Ideals is Lowest Common Multiple
- Irreducible Elements of Ring of Integers

### M

### N

- Natural Numbers are Non-Negative Integers
- Natural Numbers form Subsemiring of Integers
- Natural Numbers Set Equivalent to Ideals of Integers
- Negative of Integer
- Non-Zero Integer has Unique Positive Integer Associate
- Non-Zero Integers are Cancellable for Multiplication
- Nth Root of Integer is Integer or Irrational

### P

- Parity of Integer equals Parity of Positive Power
- 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
- Product of Absolute Values of Integers
- Properties of Integers

### Q

### R

- Rational Square Root of Integer is Integer
- Real Number lies between Unique Pair of Consecutive Integers
- Ring Epimorphism from Integers to Integers Modulo m
- Ring Monomorphism from Integers to Rationals
- Ring of Integers has Characteristic Zero
- Ring of Integers has no Zero Divisors
- Ring of Integers is Principal Ideal Domain

### 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 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
- Set of Inverse Positive Integers with Zero is Compact
- Set of Ordered Pairs of Integers is Countable Infinite
- Set of Positive Integers does not form Ring
- Strictly Increasing Infinite Sequence of Integers is Cofinal in Natural Numbers
- Strictly Increasing Infinite Sequence of Positive Integers is Cofinal in Natural Numbers
- Strictly Positive Integers have same Cardinality as Natural Numbers
- Subgroup of Integers is Ideal
- Subring of Integers is Ideal
- Subrings of Integers are Sets of Integer Multiples
- 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