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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2014 • Question 5

Question

Math 6310 • Algebra Fall 2014 • Question 5

Let \(A\) be a commutative ring and let \[ 0\longrightarrow L\longrightarrow M\longrightarrow N\longrightarrow 0 \] be a short exact sequence of \(A\)-modules. If \(L\) and \(M\) are injective, show that the sequence splits, and that \(N\) is also injective.

Raw LaTeX

Let \(A\) be a commutative ring and let
\[
0\longrightarrow L\longrightarrow M\longrightarrow N\longrightarrow 0
\]
be a short exact sequence of \(A\)-modules. If \(L\) and \(M\) are injective, show that the sequence
splits, and that \(N\) is also injective.

Topic Tags

Exact Sequences of Modules