2024-09-26

This following is a very brief summary of what happened in class on 2024-09-26.

We defined module morphisms between left $R$-modules and looked at a few examples, namely the following cases:

• $R=\mathbf{Z}$ : In this case, $\mathbf{Z}$-modules are abelian groups and morphisms of such modules are simply group morphisms.
• $R=F$, a field: In this case, $F$-modules are $F$-vector spaces and morphisms of such modules are $F$-linear transformations.
• $R=F[x]$, where $F$ is a field: In this case, $F[x]$-modules are $F$-vector spaces $V$ equipped with an $F$-linear endomorphism $T:V\to V$ (which encodes how $x$ acts on $V$). For such modules, morphisms correspond to $F$-linear transformations "compatible with" the given $F$-linear endomorphisms.

We also noted that for every ring $R$ there is a special $R$-module, called the zero module and denoted $0$, that is both "initial" and "terminal" in the category of left $R$-modules. In other words, for every $R$-module $M$ there is both a unique module morphism $0\to M$ and a unique module morphism $M\to 0$. As a consequence, for every pair of $R$-modules $M$ and $N$ there is a unique module morphism from $M$ to $N$ through $0$.

Concepts

• Module morphisms

References

• Dummit & Foote: Section 10.2