Dual of Schanuel's Lemma

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Let $0 rightarrow M rightarrow E rightarrow K rightarrow 0$ and $0
rightarrow M rightarrow E' rightarrow K' rightarrow 0$ be short
exact sequences with $M$ a left $R$-module and $E, E'$ injective left
R-modules. Prove the dual version of Schanuel’s Lemma by showing that
E ⊕ K′ $cong$ E′ ⊕ K.




Two questions:



  1. What is the dual of Schanuel's Lemma precisely?

  2. How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?



abstract-algebra modules




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  • 1




    The dual of S's lemma is $Eoplus K'cong E'oplus K$.
    – Lord Shark the Unknown
    Jul 18 at 15:58










  • What does it mean for a E, E' to be "injective left R-modules"? I thought that only functions could be injective.
    – Tomislav Ostojich
    Jul 20 at 15:24










  • en.wikipedia.org/wiki/Injective_module
    – Lord Shark the Unknown
    Jul 20 at 15:25














up vote
2
down vote

favorite













Let $0 rightarrow M rightarrow E rightarrow K rightarrow 0$ and $0
rightarrow M rightarrow E' rightarrow K' rightarrow 0$ be short
exact sequences with $M$ a left $R$-module and $E, E'$ injective left
R-modules. Prove the dual version of Schanuel’s Lemma by showing that
E ⊕ K′ $cong$ E′ ⊕ K.




Two questions:



  1. What is the dual of Schanuel's Lemma precisely?

  2. How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?



abstract-algebra modules




share|cite|improve this question

















  • 1




    The dual of S's lemma is $Eoplus K'cong E'oplus K$.
    – Lord Shark the Unknown
    Jul 18 at 15:58










  • What does it mean for a E, E' to be "injective left R-modules"? I thought that only functions could be injective.
    – Tomislav Ostojich
    Jul 20 at 15:24










  • en.wikipedia.org/wiki/Injective_module
    – Lord Shark the Unknown
    Jul 20 at 15:25












up vote
2
down vote

favorite









up vote
2
down vote

favorite












Let $0 rightarrow M rightarrow E rightarrow K rightarrow 0$ and $0
rightarrow M rightarrow E' rightarrow K' rightarrow 0$ be short
exact sequences with $M$ a left $R$-module and $E, E'$ injective left
R-modules. Prove the dual version of Schanuel’s Lemma by showing that
E ⊕ K′ $cong$ E′ ⊕ K.




Two questions:



  1. What is the dual of Schanuel's Lemma precisely?

  2. How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?



abstract-algebra modules




share|cite|improve this question














Let $0 rightarrow M rightarrow E rightarrow K rightarrow 0$ and $0
rightarrow M rightarrow E' rightarrow K' rightarrow 0$ be short
exact sequences with $M$ a left $R$-module and $E, E'$ injective left
R-modules. Prove the dual version of Schanuel’s Lemma by showing that
E ⊕ K′ $cong$ E′ ⊕ K.




Two questions:



  1. What is the dual of Schanuel's Lemma precisely?

  2. How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?




abstract-algebra modules





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 18 at 15:37









Javi

2,1631725




2,1631725









asked Jul 18 at 15:34









Tomislav Ostojich

475313




475313







  • 1




    The dual of S's lemma is $Eoplus K'cong E'oplus K$.
    – Lord Shark the Unknown
    Jul 18 at 15:58










  • What does it mean for a E, E' to be "injective left R-modules"? I thought that only functions could be injective.
    – Tomislav Ostojich
    Jul 20 at 15:24










  • en.wikipedia.org/wiki/Injective_module
    – Lord Shark the Unknown
    Jul 20 at 15:25












  • 1




    The dual of S's lemma is $Eoplus K'cong E'oplus K$.
    – Lord Shark the Unknown
    Jul 18 at 15:58










  • What does it mean for a E, E' to be "injective left R-modules"? I thought that only functions could be injective.
    – Tomislav Ostojich
    Jul 20 at 15:24










  • en.wikipedia.org/wiki/Injective_module
    – Lord Shark the Unknown
    Jul 20 at 15:25







1




1




The dual of S's lemma is $Eoplus K'cong E'oplus K$.
– Lord Shark the Unknown
Jul 18 at 15:58




The dual of S's lemma is $Eoplus K'cong E'oplus K$.
– Lord Shark the Unknown
Jul 18 at 15:58












What does it mean for a E, E' to be "injective left R-modules"? I thought that only functions could be injective.
– Tomislav Ostojich
Jul 20 at 15:24




What does it mean for a E, E' to be "injective left R-modules"? I thought that only functions could be injective.
– Tomislav Ostojich
Jul 20 at 15:24












en.wikipedia.org/wiki/Injective_module
– Lord Shark the Unknown
Jul 20 at 15:25




en.wikipedia.org/wiki/Injective_module
– Lord Shark the Unknown
Jul 20 at 15:25















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