Commutative ring

Depends on

• Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
• Binary operation on a set; associativity, commutativity, and a subset closed under the operation

Used by

• Global sections of projective twists Corollary
• Intermediate cohomology of projective twists vanishes Corollary
• Top cohomology of projective twists Corollary
• A local ring is a nonzero commutative ring with a unique maximal ideal Definition
• Algebras over a commutative ring, central structure maps, and algebra homomorphisms Definition
• An R-linear action of G on a left R-module, and a G-module over R Definition
• Balanced maps from a right module and a left module, and bilinear maps over a commutative ring Definition
• Bounded polynomial-factorization categories and the cotangent module diagram Definition
• Colourings, weight functions, and the pattern inventory Definition
• Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring Definition
• Commutative Hopf algebras over a field Definition
• Derivation of an algebra Definition
• Elementary ideals of a finitely presented module Definition
• Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose Definition
• Finite rectangular matrices over a commutative ring, their entries, rows and columns Definition
• Finite weighted directed multigraphs, weighted walks and their transfer matrices Definition
• Formal power series over a commutative ring and the coefficient-extraction functional [xⁿ] Definition
• Multiplicative subsets and the localisation S⁻¹R as equivalence classes of fractions Definition
• Noetherian commutative rings and modules Definition
• Ordered ring: a ring with a total order compatible with addition and with positives closed under multiplication Definition
• Prime ideals and maximal ideals in a commutative ring Definition
• R-orientation of a topological manifold Definition
• Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished Definition
• Simplicial objects, simplicial commutative rings and homotopy groups Definition
• Singular knots and Vassiliev derivatives Definition
• Square-zero extensions, small extensions and first-order thickenings Definition
• Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations Definition
• Symmetric polynomials as the invariants of variable permutations Definition
• The category FI and FI-modules over a commutative ring Definition
• The center of a ring Definition
• The cotangent complex of a ring map Definition
• The degenerate affine Hecke algebra Definition
• The group ring R[G] of finitely supported formal R-linear combinations of group elements Definition
• The incidence functions I(P,R) of a locally finite poset and their convolution Definition
• The Laurent polynomial ring as the principal localisation of Z[t] at t Definition
• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution Definition
• The polynomial ring R[xᵢ:i∈ I] as finitely supported coefficient families on monomials Definition
• The product ring R × S with componentwise operations, its identity (1_R, 1_S) and its units R^× × S^× Definition
• The R-module Hom_R(M,N) over a commutative ring Definition
• The radical of an ideal Definition

…and 41 more results.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Commutative ring (Wikipedia) (standard reference, not scraped)
• Thomas W. Judson, Abstract Algebra: Theory and Applications, §16.3: Rings (standard reference, not scraped)