The free module on a set and its standard basis

Depends on

• The direct sum of an indexed family of modules
• Generated submodule, cyclic and finitely generated modules, module basis and free module

Used by

• A finitely generated PID module is its torsion submodule direct-summed with a finite free module Corollary
• Over a Noetherian ring the homomorphism module between two finitely generated modules is finitely generated Corollary
• Rᵐ⊗_RRⁿ≅ Rᵐⁿ with the product basis, and dim_F(V⊗_FW)=dim_FV dim_FW Corollary
• An infinite-dimensional tensor-dual functional outside the image Counterexample
• ℤ/nℤ is generated but not free as a ℤ-module for n>1 Counterexample
• Exterior Algebra Of A Finite Free Module Definition
• Finitely presented modules and finitely presented algebras Definition
• Integral path ring of a finite quiver Definition
• Integral Specht lattice and base change Definition
• Integral tabloid form and Specht Gram matrix Definition
• Invariant basis number and the rank of a free module Definition
• Minimal Free Resolution Over A Local Ring Definition
• Temperley–Lieb link-state standard modules by defect number Definition
• Temperley–Lieb planar diagram data and endomorphism modules Definition
• The group ring R[G] of finitely supported formal R-linear combinations of group elements Definition
• The Jantzen filtration of a lifted cell module Definition
• The mod-two Morse chain group Definition
• The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums Definition
• The triangulated K₀ of the Khovanov–Seidel projective category Definition
• The trivial representation, the regular representation, and permutation representations from finite G-sets Definition
• Explicitly lifting the two simple labels in a formal rank-two algebra Example
• The free-abelian-group monad sends a set to its finite formal integer combinations Example
• The standard basis and a universal map from R³ Example
• The unit inserts basis vectors and the counit evaluates formal linear combinations in the free-vector-space adjunction Example
• ℤ/2ℤ is projective but not free over ℤ/6ℤ Example
• Every injective module is projective (refuted under the Axiom of Choice) False statement
• Every projective module is free False statement
• Flat modules need not have projective dimension zero False statement
• A nonzero PID submodule has a maximal coordinate ideal and a primitive pivot Lemma
• Central actions on nilradical cohomology factor through the Harish–Chandra projection Lemma
• Degreewise direct sums and homogeneous free covers in graded modules Lemma
• Finite direct summands over a formal-series ring are free with rank read from reduction Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• For a commutative ring, Hom_R(Rⁿ,N)≅ Nⁿ Lemma
• Group algebras, finite-group function algebras, and polynomial Hopf algebras Lemma
• Integer-valued finite formal sums of words form unital convolution rings Lemma
• Matrix units lift through a finite free split formal deformation Lemma
• Multivariate polynomial and Laurent rings over commutative rings, domains and fraction fields Lemma
• Orthogonal idempotents lift over a complete formal power-series ring Lemma
• Presentation base change and transport of explicit bases to commutative specializations Lemma

…and 29 more results.

Dependency tree · two levels

6 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

• A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
• The Stacks Project, Algebra (standard reference, not scraped)
• P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)