# Category:Ideal Theory

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This category contains results about **Ideal Theory**.

Definitions specific to this category can be found in Definitions/Ideal Theory.

**Ideal Theory** is a branch of abstract algebra which studies ideals of rings.

## Subcategories

This category has the following 18 subcategories, out of 18 total.

### A

### B

- Bézout Domains (1 P)

### C

- Coprime Ideals (1 P)

### D

### E

- Examples of Ideals of Rings (6 P)

### I

- Ideals of Rings (empty)
- Integral Ideals (4 P)

### M

### P

- Primary Ideals (4 P)

### Q

### R

- Radical of Ideals (11 P)

## Pages in category "Ideal Theory"

The following 109 pages are in this category, out of 109 total.

### C

- Canonical Injection from Ideal of External Direct Sum of Rings
- Commutative and Unitary Ring with 2 Ideals is Field
- Condition for Ideal to be Total Ring
- Conditions for Internal Ring Direct Sum
- Congruence Relation and Ideal are Equivalent
- Congruence Relation on Ring induces Ideal
- Contraction of Extension of Contraction of Ideal is Contraction

### E

### I

- Ideal Contained in Finite Union of Prime Ideals
- Ideal Contains Extension of Contraction
- Ideal induced by Congruence Relation defines that Congruence
- Ideal induces Congruence Relation on Ring
- Ideal is Additive Normal Subgroup
- Ideal is Bimodule over Ring
- Ideal is Contained in Contraction of Extension
- Ideal is Subring
- Ideal is Unit Ideal iff Includes Unity
- Ideal of External Direct Sum of Rings
- Ideal of Unit is Whole Ring
- Ideal of Unit is Whole Ring/Corollary
- Ideal Quotient is Ideal
- Ideals Containing Ideal Form Lattice
- Ideals Containing Ideal Isomorphic to Quotient Ring
- Ideals of Division Ring
- Ideals of Field
- Ideals of Ring of Integers Modulo m
- Increasing Union of Ideals is Ideal
- Increasing Union of Ideals is Ideal/Chain
- Increasing Union of Ideals is Ideal/Sequence
- Increasing Union of Sequence of Ideals is Ideal
- Integer Divisor is Equivalent to Subset of Ideal
- Integral Ideal is Ideal of Ring
- Intersection of All Ring Ideals Containing Subset is Smallest
- Intersection of Ideals of Ring contains Product
- Intersection of Integer Ideals is Lowest Common Multiple
- Intersection of Ring Ideals is Ideal
- Intersection of Ring Ideals is Largest Ideal Contained in all Ideals

### N

### P

- Polynomial Forms over Field form Integral Domain/Formulation 1
- Polynomials of Congruent Ring Elements are Congruent
- Preimage of Ideal under Ring Homomorphism is Ideal
- Preimage of Image of Ideal under Ring Homomorphism
- Prime Ideal Including Ideal Includes Radical
- Prime Number iff Generates Principal Maximal Ideal
- Principal Ideal Domain fulfills Ascending Chain Condition
- Principal Ideal Domain is Unique Factorization Domain
- Principal Ideal from Element in Center of Ring
- Principal Ideal is Ideal
- Principal Ideal is Smallest Ideal
- Principal Ideal of Commutative Ring
- Principal Ideals of Integers
- Principal Left Ideal is Left Ideal
- Principal Right Ideal is Right Ideal
- Projection onto Ideal of External Direct Sum of Rings
- Property of Being an Ideal is not Transitive

### Q

- Quotient Epimorphism from Integers by Principal Ideal
- Quotient Group of Ideal is Coset Space
- Quotient of Commutative Ring by Nilradical is Reduced
- Quotient Ring by Null Ideal
- Quotient Ring Defined by Ring Itself is Null Ring
- Quotient Ring is Ring
- Quotient Ring of Integers and Principal Ideal from Unity
- Quotient Ring of Integers with Principal Ideal

### R

### S

- Second Isomorphism Theorem for Rings
- Second Isomorphism Theorem/Rings
- Set of Ideals forms Complete Lattice
- Set of Ring Elements forming Zero Product with given Element is Ideal
- Subgroup of Additive Group Modulo m is Ideal of Ring
- Subgroup of Integers is Ideal
- Subring Generated by Unity of Ring with Unity
- Subring is not necessarily Ideal
- Subring of Integers is Ideal
- Sum of Ideals is Ideal
- Sum of Ideals is Ideal/Corollary
- Sum of Ideals is Ideal/General Result
- Sum of Ideals is Ideal/General Result/Corollary
- Sum of Integer Ideals is Greatest Common Divisor