Mixing
Dual of Schanuel's Lemma
Clash Royale CLAN TAG#URR8PPP
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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:
- What is the dual of Schanuel's Lemma precisely?
- How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?
abstract-algebra modules
edited Jul 18 at 15:37
Javi
2,1631725
asked Jul 18 at 15:34
Tomislav Ostojich
475313
add a comment |Â
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:
- What is the dual of Schanuel's Lemma precisely?
- How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?
abstract-algebra modules
edited Jul 18 at 15:37
Javi
2,1631725
asked Jul 18 at 15:34
Tomislav Ostojich
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
add a comment |Â
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:
- What is the dual of Schanuel's Lemma precisely?
- How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?
abstract-algebra modules
edited Jul 18 at 15:37
Javi
2,1631725
asked Jul 18 at 15:34
Tomislav Ostojich
475313
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:
- What is the dual of Schanuel's Lemma precisely?
- How does "showing that E ⊕ K′ $cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?
abstract-algebra modules
edited Jul 18 at 15:37
Javi
2,1631725
asked Jul 18 at 15:34
Tomislav Ostojich
475313
edited Jul 18 at 15:37
Javi
2,1631725
edited Jul 18 at 15:37
Javi
2,1631725
edited Jul 18 at 15:37
Javi
2,1631725
2,1631725
asked Jul 18 at 15:34
Tomislav Ostojich
475313
asked Jul 18 at 15:34
Tomislav Ostojich
475313
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
add a comment |Â
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
add a comment |Â
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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