Direct product module

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Be sure to mention the function construction (although indexed families are really just maps...)

Definition

Let $(R,+,*,0)$ be a ring (with out without unity), and let $(M_\alpha)_{\alpha\in I}$ be an indexed family of left $R$-modules. We can construct a new module, denoted $\prod_{\alpha\in I}M_\alpha$ that is a categorical product of the members of the family $(M_\alpha)_{\alpha\in I}$:

• $\prod_{\alpha\in I}M_\alpha$ is the underlying set of the module (we define $M:=\prod_{\alpha\in I}M_\alpha$ for convenience). This is a standard Cartesian product. The operations are:
  1. Addition: $+:M\times M\rightarrow M$ by $+:((x_\alpha)_{\alpha\in I},(y_\alpha)_{\alpha\in I})\mapsto(x_\alpha+y_\alpha)_{\alpha\in I}$ (standard componentwise operation)
  2. Multiplication/Action: $\times:R\times M\rightarrow M$ given by $\times:(r,(x_\alpha)_{\alpha\in I})\mapsto(rx_\alpha)_{\alpha\in I}$, again standard componentwise definition.

Claim 1: this is indeed an $R$-module.

With this definition we also get canonical projections, for each $\beta\in I$:

• $\pi_\beta:M\rightarrow M_\beta$ given by $\pi:(x_\alpha)_{\alpha\in I}\mapsto x_\beta$

Claim 2: the canonical projections are module homomorphisms

Claim 3: the $R$-module $M$ is the unique $R$-module such that all the projections are module homomorphisms.

Characteristic property of the direct product module

$\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}M_\alpha \ar[dd] \\ & & \\ M \ar[uurr]^\varphi \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.65){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{\varphi_c,\ \varphi_b,\ \varphi_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}$

TODO: Description

Let $(R,*,+,0)$ be a ring (with or without unity) and let $(M_\alpha)_{\alpha\in I}$ be an arbitrary indexed family of $R$-modules. Let $\prod_{\alpha\in I}M_\alpha$ be their direct product, as usual. Then:

• For any $R$-module, $M$ and
  • For any indexed family $(\varphi_\alpha:M\rightarrow M_\alpha)_{\alpha\in I}$ of module homomorphisms
    • There exists a unique morphism, $\varphi:M\rightarrow\prod_{\alpha\in I}M_\alpha$ such that:
      • $\forall\alpha\in I[\pi_\alpha\circ\varphi=\varphi_\alpha]$

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TODO: Link to diagram, this basically says it all though!

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Proof of claims

Grade: B

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The message provided is:

Do this section, or at least leave a guide, it should be routine and mirror the other instances of products

See also

• Direct sum of modules - instances of a co-product
  • External direct sum of modules
  • Internal direct sum of modules
• Characteristic property of the direct product module