2025-10-09

This following is a very brief summary of what happened in class on 2025-10-09.

We began by considering the possibility of a left adjoint to the forgetful functor $U:R\text{-}\mathbf{\text{Mod}}\to \mathbf{Set}$. First, we described the properties such an adjoint would need to have. We then carried out an analysis of what properties $F(X)$ would need to possess for some particular sets. We quickly deduced that:

• For the empty set we must have $F(\mathrm{\varnothing })\simeq 0$, the zero module
• For the singleton set we likely have $F(\{\bullet \})\simeq R$

I mentioned that we will eventually see that left adjoints commute with colimits; e.g., disjoint unions in $\mathbf{S}\mathbf{e}\mathbf{t}$, direct sums in $R\text{-}\mathbf{\text{Mod}}$. Once we know this, we can immediately deduce that for any set $X$ we must have

This inspired us to realize that only possible definition of $F(X)$ must be ${\displaystyle \underset{x\in X}{⨁}R}$. In the language of "formal sums," this means that the set of elements of $F(X)$ is the set of all finite, formal $R$-linear combinations of elements of $X$. We called $F(X)$ the free module on $X$.

We then briefly outlined:

• The $R$-module structure on $F(X)$
• The arrow function of the functor $F:\mathbf{S}\mathbf{e}\mathbf{t}\to R\text{-}\mathbf{\text{Mod}}$, i.e., how each set map $f:X\to Y$ should be sent to an $R$-module morphism $F(f):F(X)\to F(Y)$
• A set map $X\to U(F(X))$ sending each element $x$ to the "basis element" $e_x$ (which is the formal sum with coefficient $1_R$ for $x$ and $0_R$ for all other elements of $X$). These set maps are the components of the unit of the adjunction $\eta :I_{\mathbf{S}\mathbf{e}\mathbf{t}}\Rightarrow UF$
• The map that takes each module morphism $F(X)\to M$ to a set map $X\to U(M)$
• The inverse map that takes each set map $X\to U(M)$ to a module morphism $F(X)\to M$

For most of these items, we explicitly defined the maps, but then simply noted what one would need to check. (I promise that none of those things we skipped actually checking are very devious or interesting!)

We ended by defining:

• what it means for a module to be free
• the submodule generated by a subset $X\subseteq U(M)$
• what it means for a subset to generate $M$
• what it means for a module to be cyclic

Concepts

Free modules
Examples of free modules
Generators for modules and submodules

References

• Dummit & Foote, Abstract Algebra: Section 10.3