2025-10-28

This following is a very brief summary of what happened in class on 2025-10-28.

We recapped our investigation into projective and injective modules, said a bit more about injective modules, and then turned our attention to the tensor product.

We first noted that for each $(R, S)$ -bimodule $M$ and every ring $T$ , we have a functor $M \otimes_{S} - : (S, T) - Bimod \to (R, T) - Bimod$ , which we could follow with a forgetful functor to $A b$ (if we wanted to have a functor most comparable to the two hom functors we previously investigated).

We then noted (without proof) that this tensor product functor is always right exact; i.e.,

Then for each short exact sequence of $(S, T)$ -bimodules

0 \to J \overset{f}{\to} K \overset{g}{\to} L \to 0,

the corresponding sequence of $(R, T)$ -bimodules

M \otimes_{S} J \overset{1_{M} \otimes f}{\to} M \otimes_{S} K \overset{1_{M} \otimes g}{\to} M \otimes_{S} L \to 0

is exact. (Showing the surjectivity of $1_{M} \otimes g$ was actually one of the midterm questions.)

We then said an $(R, S)$ -bimodule was flat (over $T$ ) if the functor $M \otimes_{S} -$ was exact, which in light of the above property is equivalent to that functor preserving injectivity.

We ended by sketching a rough Venn diagram involving projective modules, injective modules and flat modules, with examples illustrating every possible option.

Concepts

Exact Sequences IV - Exact Sequences and Functors

References

Dummit & Foote, Abstract Algebra: Section 10.5