Notes in lecture01 - modules

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Status	Last Update	Fields
Published	10/25/2023	homomorphism of R-modules / R-linear
Published	10/25/2023	left (unitary) R-Module
Published	10/25/2023	How does every left R-module define a right R-Module?
Published	10/25/2023	An R-Module (\(μ\)) induces a ring homomorphism.
Published	10/25/2023	{{c2::A left R-module structure on M}} induces a {{c1::unital ring homomorphism \(R → End(M)\)}} and conversely
Published	10/25/2023	{{c2::A right R-module structure on M}} induces a {{c1::unital ring anti-homomorphism \(R → End(M)\)}} and conversely
Published	10/25/2023	What structure does \(End(M)\) have
Published	10/25/2023	Let M be an R-Module. Then {{c1::R is a field}} {{c3::\(⇔\)::\(⇒/⇔/⇐\)}} {{c2::M is a vector space over R}}.
Published	10/25/2023	Connection between abelian groups and \(ℤ-\)Modules
Published	10/25/2023	What \(ℤ/n\)-modules are there?
Published	10/25/2023	What \(ℤ[x]\)-modules are there
Published	10/25/2023	For any ring R, \(Hom(ℤ[x], R) \cong {{c1::R}} \)
Published	10/25/2023	direct sum of R-Modules
Published	10/25/2023	Then \(R\)-module structure on \(R/I\)
Published	10/25/2023	Are arbitrary direct sums or products of modules also modules?
Published	10/25/2023	Isomorphism suggested by the basis of a vector space
Published	10/25/2023	M is an R-Module. \(A ⊂ M\) generates M if...
Published	10/25/2023	Let M be an R-Module. \(A⊂M\) is linearly independent if...
Published	10/25/2023	basis of R-Module M
Published	10/25/2023	free module
Published	10/25/2023	Why is   \(\bigoplus_{α∈I} R \) a free 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	What isomorphism does the basis A of a free module M induce?
Published	10/25/2023	It is true for rings that \[R^m \cong R^n ⇒ n=m?\]
Published	10/25/2023	\(Hom_R(M,N)\) (M, N are R-modules)
Status	Last Update	Fields