2025-10-30

This following is a very brief summary of what happened in class on 2025-10-30.

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.

Warning!

In my original "module-focused" definition of an $R$ -algebra, I forgot one key part of the definition, namely that the $R$ -action is compatible with the internal product; i.e., that we can either multiply internally first and then act, or act on one of the two elements and then multiply the result. In other words, for every $r \in R$ and $a_{1}, a_{2} \in A$ , we must have

r (a_{1} a_{2}) = (r a_{1}) a_{2} = a_{1} (r a_{2}) .

I initially forgot that last equality, which caused us to get stuck for a bit trying to show the equivalence of this definition with the ring-focused definition. It's exactly this last equality that matches with the ring-focused condition that the ring morphism $f : R \to A$ have image contained in the center of $A$ .

Sorry again for the glitch!

We also quickly defined morphisms of algebras.

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

Concepts

Algebras

References

Dummit & Foote, Abstract Algebra: Section 11.5