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
• 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
• Invariant basis number and the rank of a free module Definition
• Minimal Free Resolution Over A Local Ring Definition
• The group ring R[G] of finitely supported formal R-linear combinations of group elements 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
• 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
• For a commutative ring, Hom_R(Rⁿ,N)≅ Nⁿ Lemma
• Integer-valued finite formal sums of words form unital convolution rings Lemma
• Subgroups of free abelian groups are free Lemma
• Submodules of finite modules over a Noetherian ring are finite by induction Lemma
• The 4m+1 path basis Lemma
• Under Choice, a submodule of an arbitrary-rank free module over a PID is free Lemma
• A submodule of an arbitrary-rank free module over a PID is free Theorem
• Euler-Poincare formula for finite free complexes Theorem
• Every nonzero commutative ring has invariant basis number for finite bases Theorem
• Flatness is equivalent to preserving injections and to the ideal and finitely generated ideal tests Theorem
• Higher Tor over the integers vanishes Theorem
• Over a Noetherian ring a module is Noetherian exactly when it is finitely generated, exactly when it is finitely presented Theorem
• The elementary tensors of two bases form the product basis of the tensor product Theorem
• The matrix category is fully faithful in modules and, with chosen bases, equivalent to finite free modules Theorem
• Universal property of the free module on a set Theorem
• Universal property of the tensor product for balanced maps into abelian groups Theorem

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)