# nov14 - Math 5125 Monday November 14 November 14 Ungraded...

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Math 5125 Monday, November 14 November 14, Ungraded Homework Exercise 10.4.4 on page 375 Show that Q Z Q and Q Q Q are isomorphic as left Q - modules. Deﬁne a Z -balanced map θ : Q × Q Q Q Q by θ ( p , q ) = p q . This induces a group homomorphism ˆ θ : Q Z Q Q Q Q satisfying ˆ θ ( p q ) = p q . It is routine to check that this is actually a Q -module map. Next note that in Q Z Q , we have px q = p xq for all x Q ; certainly this is true for all x Z , but for the general rational number x , we need some more explanation. Write x = a / b , where a , b Z and b 6 = 0. Then px q = p ( a / b ) q = p / b aqb / b = p q ( a / b ) = p xq . It now follows that we can deﬁne a Q -balanced map φ : Q × Q Q Z Q by φ ( p , q ) = p q (note that φ ( px , q ) = φ ( p , xq ) for all x Q by the previous paragraph), so φ induces a Z - module map ˆ φ Q Q Q Q Z Q satisfying ˆ φ ( p

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nov14 - Math 5125 Monday November 14 November 14 Ungraded...

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