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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2006 • Question 5

Question

Math 6310 • Algebra Spring 2006 • Question 5

Let \(R\) be a commutative ring with 1 and \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0,\qquad 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] be short exact sequences of \(R\)-modules where \(P\) and \(P'\) are projective \(R\)-modules. Show that there is an isomorphism of \(R\)-modules \(K\oplus P'\cong K'\oplus P\).

Raw LaTeX

Let \(R\) be a commutative ring with 1 and
\[
0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0,\qquad
0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0
\]
be short exact sequences of \(R\)-modules where \(P\) and \(P'\) are projective \(R\)-modules.
Show that there is an isomorphism of \(R\)-modules \(K\oplus P'\cong K'\oplus P\).

Topic Tags

Exact Sequences of Modules Projective Modules Tor Functors and Projective Resolutions