Abstract algebra questions and answers

Recent questions in Abstract algebra
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
ψ:M1××MpN
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...?

Coming up with good abstract algebra examples is essential for those who are trying to come up with the answers to theoretical questions both in Engineering and Data Science disciplines. The college students will be able to discover abstract algebra questions and answers provided by our friendly experts that will help you to understand abstract algebra questions with various examples based on high-energy physics, cryptography, and the number theory. Remember that the trick is to use number sequences to generalize the set of various integers and transformations of functions equation problems at play. Don’t forget about the application of the Algebraic number theory studies as well.