2024-10-17

This following is a very brief summary of what happened in class on 2024-10-17.

We began by observing that our construction of the tensor product $S{\otimes }_{R}M$ could evidently be extended to a construction for two modules $M$ and $N$. More specifically, we noted that if $M$ had a right $S$-action and $N$ had a left $S$-action, then we could follow our previous construction to produce (an abelian group) that presumably deserved to be denoted $M{\otimes }_{S}N$.

We then went on to note that if $M$ had a left $R$-action it would be easy to put an analogous left $R$-action on $M{\otimes }_{S}N$; and similarly if $N$ had a right $T$-action, we could use that to define a right $T$-action on $M{\otimes }_{S}N$.

These thoughts led us to extend the notion of "module" to that of a bimodule. After doing so, we went through a quick list of examples relating these new bimodules to our one-side modules. We also defined bimodule morphisms, which were as expected.

We ended by noting that our previous construction really does directly adapt to a new bimodule construction, namely that whenever we have an $(R,S)$-bimodule $M$ and $(S,T)$-bimodule $N$, we can form an $(R,T)$-bimodule that presumably deserves the name $M{\otimes }_{S}N$.

We were left with the following questions:

We'll answer both questions next time!

Concepts

• Bimodules
• Bimodule morphisms
• Tensor Products II - Tensor products of bimodules

References

• Dummit & Foote: Section 10.4