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kaliitcri 2022-02-15 Answered

Northcott Multilinear Algebra Universal Property Proof
Northcott Multilinear Algebra poses a problem. Consider R-modules M1,,Mp, M and N. Consider multilinear mapping
Northcott calls the universal problem as the problem to find M and multilinear mapping ϕ:M1××MpM such that there is exactly one R-module homomorphism h:MN such that hϕ=ψ.
If λ and λ exist I understand why the equalities at the end of the sentence follow, based on the satisfaction of the universal problem. I can't see however why homomorphisms λ and λ should exist.
I did more group theory many years ago and this is my first serious foray into "modules" so I wouldn't be surprised if there is something obvious I'm missing.
my thoughts: Clearly M and M′ are both homomorphic to N through h and h′, I'm not sure if this says anything about a relationship between M and M′ though.
If h′ were injective I could say something like λ(m)=h1(h(m)) but I don't know if there is any guarantee that h′ is injective..
Likewise, if ϕ were injective I could define λ(m)=ϕ(ϕ1(m)) but again I don't know why this would be the case...
I've tried replacing M and N with more familiar vector spaces and R-module homomorphisms by multilinear maps for better intuition but no luck.. I do know that if M and M′ are vector spaces with the same dimension then there is an isomorphism between them. I guess more generally if M and M′ have different dimensions (say dim(M)>dim(M)) then there is a homomorphism from M into a subspace of M′ and another homormophism from M′ onto M. Maybe this carries over to modules and is in the right direction for what I need...?

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