Associative rings and algebras Basic notions Concept
Ring
Summary
A ring is a set with an addition and a multiplication that obey the usual rules of arithmetic. The multiplication need not be commutative. The integers, the polynomials and the square matrices are the basic examples. The definition of a ring gives the axioms.
Where this leads
62 pages build on this one.
- Unit-regular ring
- Direct limit of rings
- Laurent polynomial ring
- Serial ring
- Triangular matrix
- Toeplitz matrix
- Gabriel–Popescu theorem
- Projective cover
- Goldie ring
- Ore localization
- Dedekind-finite ring
- Projective line over a ring
- Self-injective ring
- Category of rings
- Monoid ring
- Algebraic K-theory
- Projective dimension
- Weak dimension
- Ore condition
- Splitting lemma
- Classical ring of quotients
- Freyd–Mitchell embedding theorem
- Simple ring
- Injective hull
- Skew group ring
- Non-surjective epimorphism of algebras
- Difference algebra
- Morita equivalence
- Ore extension
- von Neumann regular ring
- Almost split sequence
- Ext functor
- Morita invariant
- Quasi-Frobenius ring
- Skew polynomial ring
- Category of modules
- Morita equivalence of a ring with its matrix rings
- Four lemma
- Progenerator
- Strassen algorithm
- Krull dimension of a noncommutative ring
- Cohn localization
- Semisimple ring
- Preadditive category
- Dyson conjecture
- Adele ring
- Semihereditary ring
- Tridiagonal matrix
- Polynomial identity ring
- Lenagan's theorem
- Projective resolution
- Hessenberg matrix
- Hereditary ring
- Group algebra
- Band matrix
- Finite ring
- Hankel matrix
- Morita duality
- Five lemma
- Invariant basis number
- Yoneda Ext and extensions
- Group ring
Further reading
- Web https://en.wikipedia.org/wiki/Ring_(mathematics), lead section
About this page
| Tag | 0B1261B5ED |
|---|---|
| URI | m2://0B1261B5ED |
| Kind | concept |
| Writing | W1 stated |
| MSC | 16B99 |
| Same as |
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What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Special cases
- Quotient ring
- Commutative ring
- Noetherian ring
- Division ring
- Subring
- Boolean ring
- Monoid ring
- Associative algebra
- Simple ring
- Partially ordered ring
- Semihereditary ring
- Ring with involution
- Hereditary ring
- Graded ring
- Polynomial identity ring
- Noncommutative ring
- Dedekind-finite ring
- Self-injective ring
- Group algebra
- Semisimple ring
- Goldie ring
- Topological ring
- Serial ring
- Finite ring
Definitions
Used by
- Direct limit of rings
- Laurent polynomial ring
- Serial ring
- Triangular matrix
- Toeplitz matrix
- Gabriel–Popescu theorem
- Projective cover
- Goldie ring
- Ore localization
- Dedekind-finite ring
- Projective line over a ring
- Self-injective ring
- Category of rings
- Monoid ring
- Algebraic K-theory
- Projective dimension
- Weak dimension
- Ore condition
- Classical ring of quotients
- Freyd–Mitchell embedding theorem
- Simple ring
- Injective hull
- Skew group ring
- Non-surjective epimorphism of algebras
- Difference algebra
- Morita equivalence
- Ore extension
- von Neumann regular ring
- Almost split sequence
- Ext functor
- Morita invariant
- Quasi-Frobenius ring
- Skew polynomial ring
- Category of modules
- Morita equivalence of a ring with its matrix rings
- Progenerator
- Strassen algorithm
- Krull dimension of a noncommutative ring
- Cohn localization
- Semisimple ring
- Preadditive category
- Adele ring
- Semihereditary ring
- Tridiagonal matrix
- Polynomial identity ring
- Lenagan's theorem
- Projective resolution
- Hessenberg matrix
- Hereditary ring
- Group algebra
- Band matrix
- Finite ring
- Hankel matrix
- Morita duality
- Invariant basis number
- Yoneda Ext and extensions
- Group ring