The category of R-algebras where R is a commutative ring is the under category R ↓ CRing. In the categorical setting we have a forgetful functor R-Alg ⥤ R-Mod. However here it extends module in order to preserve definitional equality in certain cases.
The codomain of an algebra.
Creating an algebra from a morphism in CRing.
R[X] is the generator of the category R-Alg.
The algebra of multivariate polynomials.
Creating an algebra from a subring. This is the dual of ring extension.
The multiplication in an algebra is a bilinear map.
Defining the homomorphism in the category R-Alg.
R-Alg ⥤ R-Mod
comap R S A is a type alias for A, and has an R-algebra structure defined on it when algebra R S and algebra S A.
R ⟶ S induces S-Alg ⥤ R-Alg
R ⟶ S induces S-Alg ⥤ R-Alg
A → HomR-Alg
(ι → A) → HomR-Alg
CRing ⥤ ℤ-Alg
CRing ⥤ ℤ-Alg
CRing ⥤ ℤ-Alg