Unformatted text preview: (x ) n =( xn ) = 0, and we deduce that x + y ∈ I andx ∈ I . We have so far shown that I is an abelian group under addition. Suppose r ∈ R . Then ( rx ) n = r ( xn ) = r = 0 and ( xr ) n = x ( rn ) = 0, and we conclude that rx and xr ∈ I . This completes the proof that I is an ideal of R . Exercise 10.2.8 on page 350 Let φ : M → N be an Rmodule homomorphism. Prove that φ ( Tor ( M )) ⊆ Tor ( N ) (where Tor ( M ) indicates the torsion elements of M ). Let m ∈ Tor ( M ) . Then there exists 0 6 = r ∈ R such that rm = 0. Then 0 = θ ( rm ) = r ( θ m ) , which shows that θ m ∈ Tor ( N ) as required....
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 Fall '07
 PALinnell
 Math, Algebra, Ring, Abelian group

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