This section contains definitions, theorems, lemmas, propositions, and corollaries from ring theory.
Definitions
Basic Structures
Homomorphisms
Ideals
- Ideal (left/right)
- Two-sided ideal
- Principal ideal
- Ideal generated by a subset
- Sum of ideals
- Product of ideals
- Intersection of ideals
- Quotient ring
Special Elements
Radicals and Special Ideals
Prime and Maximal Ideals
Special Rings
Ring Constructions
Polynomial Rings
- Polynomial ring
- Laurent polynomial ring
- Formal power series ring
- Content of a polynomial
- Primitive polynomial
- Irreducible polynomial
- Minimal polynomial (over a field)
Factorization Domains
- Euclidean domain
- Principal ideal domain (PID)
- Unique factorization domain (UFD)
- Prime element
- Irreducible element
- Associated elements
- Greatest common divisor
- Least common multiple
Localization
Axioms
Theorems
Isomorphism Theorems
- First isomorphism theorem (rings)
- Second isomorphism theorem (rings)
- Third isomorphism theorem (rings)
- Correspondence theorem (rings)
Fundamental Theorems
- Chinese remainder theorem
- Existence of maximal ideals (Zorn)
- Wedderburn's little theorem
- Artin–Wedderburn theorem
- Hilbert basis theorem
Algebraic Geometry
Factorization
- Gauss's lemma (content)
- Eisenstein's criterion
- Unique factorization theorem
- Euclidean domain ⇒ PID
- PID ⇒ UFD
- Gauss's theorem (UFD ⇒ polynomial ring is UFD)
Lemmas
- Gauss lemma (content multiplicativity)
- Maximal ideals are prime
- Fields are exactly commutative division rings
- Maximal ideal iff quotient is field
- Prime ideal iff quotient is integral domain
- Universal property of quotient rings
- Kernels are two-sided ideals
Propositions
- Ring homomorphisms preserve 0, 1, +, ×
- Kernel is an ideal
- Image is a subring
- Ideal correspondence
- Units map to units
- Commutative ring is field iff only ideals are (0) and (1)
- Cancellation in integral domains
- Characteristic of integral domain is 0 or prime
- UFD implies GCDs exist
- Euclidean algorithm yields gcd and Bézout identity
- Content formula
- Nilradical = intersection of prime ideals
- Idempotents ↔ product decompositions
- Chinese remainder decomposition