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

• 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
• Colourings, weight functions, and the pattern inventory Definition
• Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring 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
• 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 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 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
• The ring R^X of all functions from a set X into a ring, with pointwise operations Definition
• Zero divisor, and integral domain: a commutative ring with 1 ≠ 0 and no zero divisors Definition
• A product of two rings with 1 ≠ 0 always has zero divisors: (1,0)(0,1) = (0,0) in ℤ × ℤ, so a product of integral domains is never an integral domain Example
• For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
• ℚ and ℝ are fields, hence commutative rings, integral domains and ordered rings, all of characteristic 0 Example
• The Cauchy sequences of rationals form a commutative ring that is not an integral domain: two eventually-constant sequences with disjoint supports multiply to zero Example
• The zero ring {0}, in which 1 = 0: a commutative ring of characteristic 1 that is not a domain, not a division ring and not a field Example
• ℤ is a commutative ring and an ordered ring, the published construction being an instance of the general definitions Example
• ℤ is an integral domain of characteristic 0 whose group of units is {1,-1}, so it is not a field: 2 is nonzero and not invertible Example
• ℤ sits inside ℚ as a subring that is not a subfield, so the inverse-closure clause of the subfield definition is doing work Example
• FALSE: In every commutative ring, each nonzero element is either a unit or a zero divisor False statement
• Cancellation characterises domains: in a commutative ring with 1 ≠ 0, the implication ab = ac and a ≠ 0 imply b = c holds if and only if the ring has no zero divisors Lemma
• Every commutative division ring is a field, so "field" and "commutative division ring" name the same structures and the published definition and the ring-theoretic one agree Lemma
• Every field is a commutative ring with 1 ≠ 0; it is an integral domain, and it is a commutative division ring Lemma
• Incidence convolution is associative and distributes over pointwise addition Lemma

…and 9 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)