Direct product module

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

Definition

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

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

Claim 1: this is indeed an [ilmath]R[/ilmath]-module.

With this definition we also get canonical projections, for each [ilmath]\beta\in I[/ilmath]:

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

Claim 2: the canonical projections are module homomorphisms

Claim 3: the [ilmath]R[/ilmath]-module [ilmath]M[/ilmath] is the unique [ilmath]R[/ilmath]-module such that all the projections are module homomorphisms.

Characteristic property of the direct product module

[ilmath]\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}[/ilmath]

TODO: Description

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

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

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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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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