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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Spring 2005 • Question 2

Question

Math 6310 • Algebra Spring 2005 • Question 2

Let \(R\) be a ring with identity. Suppose \[ 0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0 \] and \[ 0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0 \] are two short exact sequences of \(R\)-modules. Suppose further that \(P\) and \(P'\) are projective. Prove that \(K'\oplus P\cong K\oplus P'\) as \(R\)-modules.

Raw LaTeX

Let \(R\) be a ring with identity. Suppose
\[
0\longrightarrow K\longrightarrow P\longrightarrow M\longrightarrow 0
\]
and
\[
0\longrightarrow K'\longrightarrow P'\longrightarrow M\longrightarrow 0
\]
are two short exact sequences of \(R\)-modules. Suppose further that \(P\) and \(P'\) are projective.
Prove that \(K'\oplus P\cong K\oplus P'\) as \(R\)-modules.

Topic Tags

Exact Sequences of Modules Ideals and Quotient Rings Projective Modules Tor Functors and Projective Resolutions