# 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 58 subcategories, out of 58 total.

### A

### B

- Boolean Rings (6 P)
- Bézout Domains (1 P)

### C

- Cyclotomic Rings (9 P)

### D

- Direct Sums of Rings (2 P)

### E

- Examples of Zero Divisors (4 P)

### F

### G

- Graded Rings (empty)
- Group Rings (1 P)

### I

- Ideals of Rings (empty)

### L

### M

### N

- Nilpotence (1 P)
- Nilpotent Ring Elements (4 P)
- Normed Rings (empty)

### O

### P

- Prime Elements of Rings (1 P)

### Q

### R

- Ring Direct Products (empty)
- Ring Endomorphisms (1 P)
- Ring Epimorphisms (13 P)
- Ring Extensions (empty)
- Ring Monomorphisms (8 P)
- Ring Zero is Unique (4 P)
- Rings of Sequences (2 P)

### S

- Semirings (2 P)
- Submultiplicative Functions (empty)

### T

- Trivial Rings (4 P)

### U

- Units of Rings (6 P)

### Z

## Pages in category "Ring Theory"

The following 87 pages are in this category, out of 87 total.

### C

### D

### E

### I

- Ideal Contains Extension of Contraction
- Ideal is Bimodule over Ring/Ring is Bimodule over Ring
- Ideal is Contained in Contraction of Extension
- 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
- Integral Multiple of Ring Element
- Integral Multiple of Ring Element/General Result
- Inverse of Central Unit of Ring is in Center
- Inverse of Unit in Centralizer of Ring is in Centralizer

### L

- 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

- Radical of Power of Prime Ideal
- Radical of Prime Ideal
- 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 by Idempotent
- Ring Homomorphism by Idempotent
- 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 Product with Zero
- Ring Subtraction equals Zero iff Elements are Equal
- Ring Zero is Idempotent
- Ring Zero is not Cancellable
- Ring Zero is Unique