Short Exact Sequences: Splitting

[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?

 

[micromass]

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

 

[jgens]

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

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!