Algebras over a commutative ring, central structure maps, and algebra homomorphisms

Depends on

• Commutative ring
• Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$

Used by

• A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• The generic point of the affine line has no relative k-valued coordinate Counterexample
• A group acting on a ring by automorphisms and its invariant subring Definition
• Associative graded algebras, bimodules, and internal shifts Definition
• Enveloping algebra and the bimodule–module dictionary Definition
• Integral elements over a commutative ring and algebraic integers Definition
• Integral elements subalgebra of an arbitrary ring map Definition
• Subalgebra generated by a subset, algebras of finite type, and module-finite algebras Definition
• A nonsplit simple has Hom-pairing diagonal two Example
• An algebra that is finite dimensional as a vector space over a field is a Noetherian ring Example
• The dual numbers have Cartan map multiplication by two Example
• The functor of points of the affine line Example
• The graded dual numbers have Cartan polynomial 1+v² Example
• A subring that admits a module retraction from a Noetherian ring is Noetherian Lemma
• Module finiteness is transitive along a tower of algebras Lemma
• Projective representations and twisted algebra modules Lemma
• Degree-zero Hochschild homology is bimodule coinvariants Proposition
• Hom_G(V,W) is a k-vector space and End_G(V) is a k-algebra Proposition
• A module-finite algebra over a Noetherian ring is a Noetherian ring, and so is every ring between the two Theorem
• Projective and simple classes are dual bases under splitting Theorem
• Projective Hom pairing descends and is graded sesquilinear Theorem
• The group ring R[G] is a unital R-algebra with basis G, and each g∈ G is a unit of R[G] Theorem
• The tensor product of R-algebras has multiplication (a⊗ b)(a'⊗ b')=aa'⊗ bb' Theorem
• Universal property of the tensor algebra Theorem

Dependency tree · two levels

7 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

• W. Li, Commutative Algebra, Lectures 9-10 (standard reference, not scraped)