2024-11-04

This following is a very brief summary of what happened in class on 2024-11-04.

We began by asking whether an $R$-module $M$ can be endowed with a multiplication to give it a full ring structure, in a way compatible with the given $R$ action. That led us to the idea of an algebra. We gave two possible definitions of an $R$-algebra, one from the point of view of modules and one from the point of view of rings. We spent most of the class verifying these two definitions are indeed equivalent.

We also quickly defined morphisms of algebras, and gave a handful of examples.

Next time we'll define a functor $\mathcal{T}:R\mathbf{\text{-Mod}}\to R\mathbf{\text{-Alg}}$ that is adjoint to the corresponding forgetful functor.

Concepts

• Algebras

References

• Dummit & Foote: Section 11.5