Linked Questions

3 votes
1 answer
499 views

If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is exact and $B\simeq A\oplus C$ as a $R$-module, does this sequence split? [duplicate]

Suppose $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence that $B\simeq A\oplus C$ as a $R$-module. Does this short exact sequence split? I think the answer is no, ...
kpax's user avatar
  • 2,921
2 votes
0 answers
85 views

Splitting of an exact sequence of abelian groups [duplicate]

Let $0\to A\xrightarrow{f} A\oplus B\xrightarrow{g}B\to 0$ be a short exact sequence of (finite) abelian groups, then must it be split exact? If not, can you show me some examples? Thanks!
LipCaty's user avatar
  • 379
13 votes
1 answer
3k views

Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

Recently, someone stated that every short exact sequence (of, say, modules) of the form $$0 → M → M \oplus N → N → 0$$ splits. I think this is false in general because the arrow $M → M \oplus N$ might ...
k.stm's user avatar
  • 18.5k
2 votes
1 answer
1k views

When does a short exact sequence of groups imply it is isomorphic to direct product group [closed]

Suppose that $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$ is a short exact sequence of groups. Then, what is a (necessary and )sufficient condition for $G\cong N\times Q$. In other words, ...
user682141's user avatar
6 votes
1 answer
1k views

A short exact sequence with $M=M_1 \oplus M_2$ that does not split

A sequence of $R$-modules of the form $$0 \to M_1 \stackrel{f}{\to} M \stackrel{g}{\to} M_2 \to 0$$ is called a short exact sequence (ses) if $f$ is injective, $g$ is surjective and $\operatorname{...
Sayan's user avatar
  • 2,688
3 votes
1 answer
1k views

Does exact sequence of abelian groups split when middle group has a subgroup direct sum of other groups?

If I have an exact sequence of abelian groups, the sequence coming from knowing that $H\cong G/F$, $$0\rightarrow F \rightarrow G \rightarrow H \rightarrow 0$$ where I know that $ F\oplus H\subset G$...
Felix Y.'s user avatar
  • 673
2 votes
1 answer
901 views

A question about split short exact sequence of modules

Let $0\longrightarrow A\stackrel{f}{\longrightarrow} C\stackrel{g}{\longrightarrow} B\longrightarrow 0$ be a split short exact sequence of modules. That is, there exists $\alpha:C\to A$ such that $\...
bfhaha's user avatar
  • 3,741
0 votes
1 answer
785 views

non-split short exact sequence $A\rightarrow B \rightarrow C$ with $B \cong A\oplus B$

From this problem here, it says that any counter example can not be finitely generated Abelian groups. But I can not find the mistake in the following counter example, I must missed something obvious, ...
Xiao's user avatar
  • 9,566
2 votes
1 answer
576 views

group extension properties: split, central, cocycle or not

Given a group extension $$0 \to A \to E \to G \to 1 $$ with $A$ an abelian group, there are several properties to describe this extension: central or non-central extension split or not trivial or ...
annie marie cœur's user avatar
3 votes
1 answer
758 views

In $R$-Mod Category, example for $B\cong A \oplus C \nRightarrow 0 \to A \to B \to C \to0$ splits.

From splitting lemma, we know in $R$-Mod Category, short exact sequence $0 \to A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C \to0$ splits if it satisfies one of the following equivalent ...
Andrews's user avatar
  • 3,961
5 votes
1 answer
388 views

$0\to C'\to C\to C''\to0$ splits if $C\cong C'\oplus C''$ as a chain complex?

Question Given a unitary ring $A$ and an exact sequence $$0\to C'\xrightarrow iC\xrightarrow pC''\to0$$ in the Abelian category of chain complexes over $A$, where $C,C',C''$ are chain complexes of ...
Yai0Phah's user avatar
  • 9,733
1 vote
1 answer
94 views

Does an embedding necessarily make a short exact sequence split?

Suppose we have a short exact sequence. $0 \to A \to B \to C \to 0$. Suppose also that we know there is some embedding $e : C \hookrightarrow B$. Can we conclude that the sequence splits? It seems ...
CuriousKid7's user avatar
  • 4,144
0 votes
0 answers
200 views

Just That Middle Term is a Direct Sum of Extremes Does Not Mean the Sequence is Exact

Does there exist a short exact sequence of the form $$0\to \mathbf Z\to \mathbf Z\oplus (\mathbf Z/n\mathbf Z) \to \mathbf Z/n\mathbf Z\to 0$$ which does not split? (Note: Don Alejo provided a link ...
caffeinemachine's user avatar
2 votes
1 answer
142 views

Split exact sequence equivalent conditions

For a short exact sequence $0 \longrightarrow A \overset{i}{\longrightarrow} B\overset{j}{\longrightarrow} C \longrightarrow 0$ of abelian groups the following statements are equivalent: There is a ...
Divider's user avatar
  • 69
0 votes
1 answer
55 views

exact sequence $A\to B\to C$ with $B\cong A\oplus C$

Let $A\xrightarrow{f} B\xrightarrow{g} C$ be an exact sequence of finitely generated abelian groups with $B\cong A\oplus C$. Question: Is the sequence short exact? If $B$ is finite, the answer is ...
LipCaty's user avatar
  • 379

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