Mon Jan 29

The category R-Mod of left R-modules and R-linear maps.
Z-Mod = abelian groups.
C-Mod = complex vector spaces.
C[x]-Mod = complex vector spaces equipped with specific endomorphisms.
Isomorphism classes in this last category correspond to Jordan canonical forms.

Submodules, quotient modules. Infinite products vs. infinite direct sums.
Left ideals.
Given m in an R-module M, we get an R-linear map R -> M, whose kernel is a left ideal.
Conversely, given a left ideal I, we get a map R -> R/I of R-modules (not rings).

I claimed that R is a (R x R^op)-module. Not so. It is a left R-module, and a left R^op-module, but we can't glue them together so easily; this will await tensor products.