Categorical notes on Modules

Products and coproducts

As in $\text{Ab}$, products and coproducts exist, and finite products and coproducts coincide in $R\text{-Mod}$. If $M$ and $N$ are $R$-modules, we can give an $R$-module structure to $M\oplus N$ (the direct sum of the underlying abelian groups) by prescribing

$$\begin{array}{rlr}r(m,n):=(rm,rn)& & \forall r\in R.\end{array}$$

Note that $M\oplus N$ comes together with several $R$-module homomorphisms:

$$\begin{array}{rl}& {\pi }_M:M\oplus N\to M,{\pi }_N:M\oplus N\to N\\ & i_M:M\to M\oplus N,i_N:N\to M\oplus N.\end{array}$$

Proposition 371.1(Aluffi (2009) III.6.1).

$M\oplus N$ satisfies the universal properties of both the product and the coproduct of $M$ and $N$.

Since $R$-module homomorphisms are in particular abelian group homomorphisms and $M\oplus N$ (as an abelian group) satisfies the universal properties of products and coproducts in $\text{Ab}$, we only need to verify that the unique map supplied by the universal properties in $\text{Ab}$ respects the $R$-module structure (Def 108.4.2); this is a trivial check.

Remark 371.2(Remark).

Checking weather a module $M$ can be decomposed as a direct sum $N\oplus L$ for submodules $N,L\subseteq M$ is equivalent to checking if the unique homomorphism $\phi :N\oplus L\to M$ given by the universal property (with the functions from $N$ and $L$ taken to be inclusions, of course) is an isomorphism. It is easily seen that this is equivalent to the conditions $M=L+N$ (surjectivity) and $L\cap N=0$ (injectivity).

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Remark 371.3(Remark).

The infinite case mirrors the situation in $\text{Ab}$:

1. The product ${\prod }_{i\in I}{M}_i$ of $R$-modules consists of all tuples $(m_i{)}_{i\in I}$ with componentwise addition and scalar multiplication.
2. The coproduct ${⨁}_{i\in I}{M}_i$ is the submodule of the product consisting of tuples with finite support.

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Kernels and Cokernels

Recall that monomorphisms and epimorphisms do not automatically satisfy good properties. They do in $R\text{-Mod}$.

Universal properties for kernels and cokernels in $R\text{-Mod}$ are analogous to the corresponding properties for groups (Prp 31.3, Def 31.5).

Proposition 371.4(Proposition).

The following hold in $R$-mod:

1. Kernels and cokernels exist;
2. $\phi$ is a monomorphism $⟺$ $\ker \phi$ is trivial $⟺$ $\phi$ is injective as a set-function;
3. $\phi$ is an epimorphism $⟺$ $\text{coker}\phi$ is trivial $⟺$ $\phi$ is surjective as a set-function.

It is easy to check that the usual definitions of kernel and cokernel satisfy their universal properties. Compare 2 and 3 with Prp 31.4, Prp 31.6; proofs are easy generalizations¹.

Remark 371.5(Remark).

The facts that $R\text{-Mod}$ has a zero object (the 0-module), its $\text{Hom}$ sets are abelian groups (with addition defined pointwise), and it has (finite) products and coproducts make $R\text{-Mod}$ an additive category. The fact that $R\text{-Mod}$ has kernels and cokernels upgrades to the status of abelian category.

Footnotes

1. Recall that while epimorphism $⟺$ surjective is true in $\text{Gp}$, Prp 31.6’s method of proof (using cokernels) only works for $\text{Ab}$ since the ‘standard’ definition of cokernel is not valid in $\text{Gp}$. We have no such hitches in $R\text{-Mod}$. ↩

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References

Aluffi, P. (2009). Algebra: Chapter 0. American Mathematical Society.