Category:Ring Theory

C

• [+] Commutative Algebra‎ (1 C, 30 P)
• [+] Commutative Rings‎ (1 C, 7 P)

D

• [+] Division Rings‎ (2 C, 11 P)

E

• [×] Euclidean Domains‎ (10 P)

F

• [+] Field Theory‎ (12 C, 43 P)
• [×] Finite Rings‎ (5 P)

I

• [+] Ideal Theory‎ (1 C, 73 P)
• [+] Integral Domains‎ (4 C, 57 P)

L

• [×] Local Rings‎ (1 P)

M

• [×] Modulo Arithmetic‎ (66 P)

P

• [+] Polynomial Theory‎ (1 C, 57 P)

Q

• [×] Quotient Rings‎ (25 P)

R

• [×] Ring Automorphisms‎ (3 P)
• [×] Ring Direct Sums‎ (1 P)
• [×] Ring Epimorphisms‎ (14 P)
• [+] Ring Homomorphisms‎ (5 C, 10 P)

R cont.

• [+] Ring Isomorphisms‎ (2 C, 12 P)
• [×] Ring Monomorphisms‎ (6 P)
• [×] Rings with Unity‎ (18 P)

S

• [×] Semirings‎ (2 P)
• [+] Subrings‎ (1 C, 28 P)

U

• [×] Unique Factorization Domains‎ (5 P)

A

• Annihilator of Ring Always Contains Zero

B

• Binomial Theorem/Integral Index
• Binomial Theorem/Ring Theory

C

• Center of Ring is Commutative Subring
• Centralizer of Ring Subset is Subring
• Characteristic of Null Ring is One
• Complex Numbers form Ring
• Congruence Relation on Ring Induces Ring

D

• Definition of Polynomial from Polynomial Ring
• Diagonal Relation on Ring is Ordering Compatible with Ring Structure

E

• Epimorphism from Division Ring to Ring
• Epimorphism Preserves Rings
• Equivalence of Definitions of Noetherian Ring
• External Direct Product of Rings

G

• Generator of Additive Group Modulo m iff Unit of Ring

I

• Idempotent Elements of Ring with No Zero Divisors
• Identity Mapping is Ordered Ring Automorphism
• Integers form Totally Ordered Ring

I cont.

• Integral Multiple Distributes over Ring Addition
• Inverse of Ordered Ring Isomorphism is Ordered Ring Isomorphism

N

• Negative of Ring Negative
• Negative Product
• Null Ring iff Zero and Unity Coincide
• Null Ring is Ring
• Null Ring is Trivial Ring

O

• Order of Squares in Totally Ordered Ring

P

• P-adic Valuation is Valuation
• P-adic Valuation on Integers
• Positive Elements of Ordered Ring
• Powers of Ring Elements
• Powers of Ring Elements/General Result
• Product is Zero Divisor means Zero Divisor
• Properties of Ordered Ring

R

• Rational Numbers form Ring
• Real Numbers form Ring
• Ring Distributand Abelian Group

R cont.

• Ring Less Zero Semigroup
• Ring Negatives are Unique
• Ring Not Empty
• Ring of Quaternions
• Ring of Square Matrices over Ring
• Ring Product with Zero
• Ring Zero is Unique
• Ring Zero Not Cancellable

S

• Structure Induced by Ring Operations is a 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

T

• Trivial Ring from Abelian Group
• Trivial Ring is Commutative Ring

V

• Valuation Ring is Local

Z

• Zero Divisor Not Cancellable
• Zero Divisor Product
• Zero Divisor Product is Zero Divisor