Algebras over commutative semirings #

In this file we define associative unital Algebras over commutative (semi)rings.

algebra homomorphisms AlgHom are defined in Mathlib.Algebra.Algebra.Hom;
algebra equivalences AlgEquiv are defined in Mathlib.Algebra.Algebra.Equiv;
Subalgebras are defined in Mathlib.Algebra.Algebra.Subalgebra;
The category AlgebraCat R of R-algebras is defined in the file Mathlib.Algebra.Category.Algebra.Basic.

See the implementation notes for remarks about non-associative and non-unital algebras.

Main definitions: #

Algebra R A: the algebra typeclass.
algebraMap R A : R →+* A: the canonical map from R to A, as a RingHom. This is the preferred spelling of this map, it is also available as:
- Algebra.linearMap R A : R →ₗ[R] A, a LinearMap.
- algebra.ofId R A : R →ₐ[R] A, an AlgHom (defined in a later file).
Instances of Algebra in this file:
- Algebra.id
- algebraNat
- algebraInt
- algebraRat
- Module.End.instAlgebra

Implementation notes #

Given a commutative (semi)ring R, there are two ways to define an R-algebra structure on a (possibly noncommutative) (semi)ring A:

By endowing A with a morphism of rings R →+* A denoted algebraMap R A which lands in the center of A.
By requiring A be an R-module such that the action associates and commutes with multiplication as r • (a₁ * a₂) = (r • a₁) * a₂ = a₁ * (r • a₂).

We define Algebra R A in a way that subsumes both definitions, by extending SMul R A and requiring that this scalar action r • x must agree with left multiplication by the image of the structure morphism algebraMap R A r * x.

As a result, there are two ways to talk about an R-algebra A when A is a semiring:

variables [CommSemiring R] [Semiring A]
variables [Algebra R A]

variables [CommSemiring R] [Semiring A]
variables [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]

The first approach implies the second via typeclass search; so any lemma stated with the second set of arguments will automatically apply to the first set. Typeclass search does not know that the second approach implies the first, but this can be shown with:

example {R A : Type*} [CommSemiring R] [Semiring A]
  [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] : Algebra R A :=
Algebra.ofModule smul_mul_assoc mul_smul_comm

The advantage of the first approach is that algebraMap R A is available, and AlgHom R A B and Subalgebra R A can be used. For concrete R and A, algebraMap R A is often definitionally convenient.

The advantage of the second approach is that CommSemiring R, Semiring A, and Module R A can all be relaxed independently; for instance, this allows us to:

Replace Semiring A with NonUnitalNonAssocSemiring A in order to describe non-unital and/or non-associative algebras.
Replace CommSemiring R and Module R A with CommGroup R' and DistribMulAction R' A, which when R' = Rˣ lets us talk about the "algebra-like" action of Rˣ on an R-algebra A.

While AlgHom R A B cannot be used in the second approach, NonUnitalAlgHom R A B still can.

You should always use the first approach when working with associative unital algebras, and mimic the second approach only when you need to weaken a condition on either R or A.