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Tuesday, July 2, 2013

Rings Inducing Projective and Injective Modules (10.5.6)

Dummit and Foote Abstract Algebra, section 10.5, exercise 6:

MathJax TeX Test Page Prove every R-module is projective if and only if every R-module is injective.

Proof: Every R-module being projective implies every short exact sequence 0DMN0 splits since N is an R-module and thus projective, therefore arbitrary D is also injective. Likewise, if every R-module is injective, then for an arbitrary R-module D we have 0LMD0 splits as L is injective, so that D is projective. 

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