Notes in lecture02 - quotient modules

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Status	Last Update	Fields
Published	10/25/2023	When is a subset of R an R-Module
Published	10/25/2023	If M {{c2::is a free R-module of rank (number of elements in basis) \(n < ∞\)}}, then {{c1::\(M \cong R^n\)}}
Published	10/25/2023	Submodule criterion
Published	10/25/2023	R-submodule of \(M\)
Published	10/25/2023	What do the submodules of \(R\) as an \(R\)-module look like
Published	10/25/2023	How to find submodules using a homomorphism
Published	10/25/2023	If \(M\) is an \(R\)-module and \(N\) is a {{c1::submodule}}, then the quotient \(M/N\) has an \(R-module\) structure given b…
Published	10/25/2023	First Isomorphism theorem for modules
Published	10/25/2023	Suppose \(N⊂M\) is a submodule. What is a good way to think about \(M/N\)?
Published	10/25/2023	submodule generated by a set
Published	10/25/2023	characterizatio of \(<A>\)
Published	10/25/2023	The R-module homomorphism \(α:N→N_1\) induces a homomorphism \(α_*\)...
Published	10/25/2023	The R-module homomorphism \(α:M→M_1\) induces a homomorphism \(α^*\)...
Published	10/25/2023	We know that for modules M,N \(Hom_R(M,N)\) is an abelian group. Describe three ways that \(Hom_R(M,N)\) could have a module structure …
Published	10/25/2023	R-S-bimodule
Published	10/25/2023	R-bimodule
Published	10/25/2023	example: bimodules over subsets of R
Published	10/25/2023	If \(M\) {{c1::is a left R-module}} and N is {{c1::a R-S-bimodule}}, then \(Hom_{R(left)}(M,N)\) is {{c2::a right S-module}} by the formula
Published	10/25/2023	If \(M\) is {{c1::a R-S-bimodule}} and N is {{c1::a left R-module}}, then \(Hom_{R(left)}(M,N)\) is {{c2::a left S-module}} by the formula
Published	10/25/2023	bilinear map of modules
Published	10/25/2023	Is a bilinear map R-linear?
Published	10/25/2023	Universal property of free modules
Published	10/25/2023	Construction of the tensor product of R-modules M,N (sketch)1. Take a free module with basis \(M×N\), that is \[F = \bigoplus_{M×N}R\]2. Tak…
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