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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 1994 • Question 6

Question

Math 6310 • Algebra Fall 1994 • Question 6

Show that if \[ 0\to K\to P\to M\to 0 \] and \[ 0\to K'\to P'\to M\to 0 \] are short exact sequences of \(R\)-modules, where \(P\) and \(P'\) are projective, then \[ K\oplus P'\cong K'\oplus P. \]

Raw LaTeX

Show that if
\[
0\to K\to P\to M\to 0
\]
and
\[
0\to K'\to P'\to M\to 0
\]
are short exact sequences of \(R\)-modules, where \(P\) and \(P'\) are projective, then
\[
K\oplus P'\cong K'\oplus P.
\]

Topic Tags

Exact Sequences of Modules Projective Modules Tor Functors and Projective Resolutions