# Short Exact Sequences: Splitting

1. Mar 29, 2012

### jgens

In Dummit and Foote, a short exact sequence of R-modules $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ ($\psi:A \rightarrow B$ and $\phi:B \rightarrow C$) is said to split if there is an R-module complement to $\psi(A)$ in $B$. The authors are not really clear on what the phrase "an R-module complement to" means, so I was hoping someone here could help clear my confusion up. My best guess is that they mean "there exists some R-module $C'$ such that $B$ is the internal direct sum of $\psi(A)$ and $C'$."

In Dummit and Foote, a short exact sequence of groups $1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1$ ($\psi:A \rightarrow B$ and $\phi:B \rightarrow C$) is said to split if there is a subgroup complement to $\psi(A)$ in $B$. Again the authors are not really clear on what the phrase "a subgroup complement to" means, so I was hoping someone here could help clear my confusion up. My best guess is that they mean "there exists some group $C'$ such that $B$ is the internal semi-direct product of $\psi(A)$ and $C'$."

Are my interpretations correct?

Last edited: Mar 29, 2012
2. Mar 29, 2012

### micromass

The relevant definitions are on page 180 and page 383 (which is not so clear, but you get the idea).

3. Mar 29, 2012

### jgens

The definition on page 180 is clear and it appears that my interpretation was correct for groups. Since R-modules are an abelian category, I am pretty sure I got the right interpretation for modules too. Thanks!