Скачать или смотреть Homological algebra 6: Injective modules

This lecture is part of an online course on commutative algebra, following the book
"Commutative algebra with a view toward algebraic geometry" by David Eisenbud.

We study to following problem: are there enough injective modules, in the sense that every module is a submodule of an injective module?

Over the integers we show that injective modules are divisible, and use this to show there are enough injective Z-modules. Then we use this to show that three are enough injective R-modules over any ring R.

We then show that every module M has a minimal injective module containing it, unique up to isomorphism. This can be characterized as an injective module that is an essential extension of M. (An essential extension N of M is a module N containing M such that every nonzero submodule of N has nonzero intersection with M.)


Reading: Section
Exercises: