# Category:Integral Domains

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This category contains results about **Integral Domains**.

Definitions specific to this category can be found in Definitions/Integral Domains.

An **integral domain** $\struct {D, +, \circ}$ is:

- a commutative ring which is non-null
- with a unity
- in which there are no (proper) zero divisors, that is:
- $\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$

that is, in which all non-zero elements are cancellable.

## Subcategories

This category has the following 21 subcategories, out of 21 total.

### A

### C

- Common Divisors (empty)

### D

### E

- Euclidean Valuations (empty)

### F

- Field is Integral Domain (3 P)

### G

### I

### O

### P

### U

- Unity Divides All Elements (3 P)

### V

- Valuation Rings (1 P)

## Pages in category "Integral Domains"

The following 54 pages are in this category, out of 54 total.

### C

### D

### E

- Element in Integral Domain is Divisor iff Principal Ideal is Superset
- Element in Integral Domain is Unit iff Principal Ideal is Whole Domain
- Element of Integral Domain Divides Zero
- Element of Integral Domain is Divisor of Itself
- Equal Elements of Field of Quotients
- Equivalence of Definitions of Associate in Integral Domain
- Equivalence of Definitions of Integral Domain
- Even Integers do not form Integral Domain
- Existence of Field of Quotients

### F

### I

### O

### P

- Prime Element of Integral Domain is Irreducible
- Prime Ideal iff Quotient Ring is Integral Domain
- Principal Ideal Domain is Unique Factorization Domain
- Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power
- Principal Ideals in Integral Domain
- Product of Units of Integral Domain with Finite Number of Units

### Q

### R

- Rational Numbers with Denominator Power of Two form Integral Domain
- Ring Homomorphism from Ring with Unity to Integral Domain Preserves Unity
- Ring of Integers Modulo Composite is not Integral Domain
- Ring of Integers Modulo Prime is Integral Domain
- Ring of Polynomial Forms is Integral Domain
- Ring of Polynomial Forms over Integral Domain is Integral Domain