nov30 - Math 5125 Wednesday, November 30 November 30,...

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Math 5125 Wednesday, November 30 November 30, Ungraded Homework Exercise 10.5.4 on page 403 Let R be a ring with a 1 and let Q 1 and Q 2 be R -modules. Prove that Q 1 Q 2 is an injective R -module if and only if Q 1 and Q 2 are injective. Suppose Q 1 Q 2 is an injective R -module; we will prove that Q 1 must also be injective. Suppose we are given a diagram of R -modules and R -maps with exact row 0 ---→ A α ---→ B y θ Q 1 We want to find an R -map φ : B Q 1 making the diagram commute, that is φα = θ . Define θ 1 : A Q 1 Q 2 by θ 1 ( a ) = ( θ a , 0 ) . Then clearly θ 1 is an R -map, so by injectivity of Q 1 Q 2 , there exists an R -map φ 1 : B Q 1 Q 2 such that αφ 1 = θ 1 . Let π 1 : Q 1 Q 2 Q 1 denote the natural projection, so π 1 ( q ) = ( q , 0 ) , and define φ = π 1 φ 1 : B Q 1 . Then φα ( a ) = π 1 φ 1 α ( a ) = π 1 θ 1 ( a ) = π 1 ( θ a , 0 ) = θ ( a ) , so φα = θ and we have shown that Q 1 is injective. Similarly
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This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

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nov30 - Math 5125 Wednesday, November 30 November 30,...

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