# ahw11 - Math 5125 Monday, November 21 Eleventh Homework...

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Math 5125 Monday, November 21 Eleventh Homework Solutions 1. Section 10.3, Exercise 7 on page 356. Let R be a ring with a 1, let M be a left R - module, and let N be a submodule of M . Prove that if both M / N and N are ﬁnitely generated, then so is M . Since M / N is ﬁnitely generated, we may write M / N = R ( x 1 + N )+ R ( x 2 + N )+ ··· + R ( x m + N ) for some m Z + and x i M , and since N is ﬁnitely generated, we may write N = Ry 1 + Ry 2 + ··· + Ry n for some n Z + and y i N . If m M , then we may write m + N = r 1 ( x 1 + N )+ ··· + r m ( x m + N ) for some r i R . This means that m - r 1 x 1 -···- r m x m N , so we may write m - r 1 x 1 -···- r m x m = s 1 y 1 + ··· + s n y n for some y i M . Then m = r 1 x 1 + ··· + r m x m + s 1 y 1 + ··· + s n y n and it follows that M is ﬁnitely generated by { x 1 , . . . , x m , y 1 , . . . , y n } . 2. Section 10.3, Exercise 13 on page 356. Let R be a commutative ring and let F be a free R -module of ﬁnite rank. Prove the following isomorphism of R -modules: Hom R ( F , R ) = F

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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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ahw11 - Math 5125 Monday, November 21 Eleventh Homework...

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