# Category:Ring Theory

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This category contains results about Ring Theory.

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

**Ring Theory** is a branch of abstract algebra which studies rings and other related algebraic structures.

## Subcategories

This category has the following 45 subcategories, out of 45 total.

### A

### B

### C

### D

### E

### F

### G

### I

### L

### M

### N

### O

### P

### Q

### R

### S

### T

### U

### Z

## Pages in category "Ring Theory"

The following 93 pages are in this category, out of 93 total.

### C

### D

### E

### I

- Ideal is Bimodule over Ring/Ring is Bimodule over Ring
- Idempotent Elements of Ring with No Proper Zero Divisors
- Increasing Union of Ideals is Ideal
- Increasing Union of Ideals is Ideal/Chain
- Integral Domain is Reduced Ring
- Integral Multiple Distributes over Ring Addition
- Inverse of Central Unit of Ring is in Center
- Inverse of Unit in Centralizer of Ring is in Centralizer

### L

- Left Ideal is Left Module over Ring/Ring is Left Module over Ring
- Left Module Does Not Necessarily Induce Right Module over Ring
- Left Module induces Right Module over same Ring iff Actions are Commutative
- Left Module over Commutative Ring induces Bimodule
- Left Module over Commutative Ring induces Right Module
- Left Module over Ring Induces Right Module over Opposite Ring

### N

### P

### R

- Rational Numbers form Ring
- Real Numbers form Ring
- Right Ideal is Right Module over Ring/Ring is Right Module over Ring
- Right Module Does Not Necessarily Induce Left Module over Ring
- Right Module induces Left Module over same Ring iff Actions are Commutative
- Right Module over Commutative Ring induces Bimodule
- Right Module over Commutative Ring induces Left Module
- Right Module over Ring Induces Left Module over Opposite Ring
- Ring Element is Zero Divisor iff not Cancellable
- Ring is Bimodule over Ring
- Ring is Left Module over Ring
- Ring is not Empty
- Ring is Right Module over Ring
- Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors
- Ring Negative is Unique
- Ring of Idempotents is Idempotent Ring
- Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring
- Ring of Quaternions is Ring
- Ring Product with Zero
- Ring Subtraction equals Zero iff Elements are Equal
- Ring with Multiplicative Norm has No Proper Zero Divisors
- Ring Zero is Idempotent
- Ring Zero is not Cancellable
- Ring Zero is Unique

### S

- Strictly Positive Integer Power Function Strictly Succeeds Each Element
- Structure Induced by Commutative Ring Operations is Commutative Ring
- Structure Induced by Ring Operations is Ring
- Structure Induced by Ring with Unity Operations is Ring with Unity
- Subtraction of Subring is Subtraction of Ring
- Sum of All Ring Products is Additive Subgroup
- Sum of All Ring Products is Associative
- Sum of All Ring Products is Closed under Addition

### U

- Union of Chain of Ideals is Ideal
- Union of Chain of Proper Ideals is Proper Ideal
- Unit of Ring of Mappings iff Image is Subset of Ring Units
- Unit of Ring of Mappings iff Image is Subset of Ring Units/Image is Subset of Ring Units implies Unit of Ring of Mappings
- Unit of Ring of Mappings iff Image is Subset of Ring Units/Unit of Ring of Mappings implies Image is Subset of Ring Units
- Unity is Unity in Ring of Idempotents
- Unity of Ring is Idempotent