Direct product module
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Be sure to mention the function construction (although indexed families are really just maps...)
Contents
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]^{[1]}:
- [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^{[Note 1]}. The operations are:
- Addition: ^{[Note 2]} [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)
- 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.^{[1]}
With this definition we also get canonical projections, for each [ilmath]\beta\in I[/ilmath]^{[1]}:
- [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^{[1]}
- 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.^{[1]}
Characteristic property of the direct product module
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^{[1]}:- 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^{[Note 3]}, [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]
- There exists a unique morphism^{[Note 3]}, [ilmath]\varphi:M\rightarrow\prod_{\alpha\in I}M_\alpha[/ilmath] such that:
- For any indexed family [ilmath](\varphi_\alpha:M\rightarrow M_\alpha)_{\alpha\in I} [/ilmath] of module homomorphisms
TODO: Link to diagram, this basically says it all though!
Proof of claims
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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
- Characteristic property of the direct product module
Notes
- ↑ An alternate construction is that [ilmath]\prod_{\alpha\in I}M_\alpha[/ilmath] consists of mappings, [ilmath]f:I\rightarrow\bigcup_{\alpha\in I}M_\alpha[/ilmath] where [ilmath]\forall\alpha\in I[f(\alpha)\in M_\alpha][/ilmath] this is just extra work as you should already be familiar with considering tuples as mappings.
- ↑ the operation of the Abelian group that makes up a module
- ↑ Morphism - short for homomorphisms in the relevant category, in this case modules
References
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} Abstract Algebra - Pierre Antoine Grillet