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# ahw3 - Math 5126 Monday February 4 Third Homework Solutions...

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Math 5126 Monday, February 4 Third Homework Solutions 1. First we write ( x + 1 )( x 3 + 1 ) as a product of irreducibles, namely ( x + 1 ) 2 ( x 2 - x + 1 ) ; it is easy to see that x 2 - x + 1 is irreducible over Q (note that the roots are not real). If R is a PID and x = p e 1 1 ... p e n n where the p i are nonassociate primes, then a ﬁnitely generated R / ( x ) -module is isomorphic to a direct sum of modules of the form R / ( p f 1 1 ... p f n n ) , where f i e i , and such a module is projective if and only if f i = e i for all i . It follows that the ﬁnitely generated projective Q [ x ] / ( x + 1 )( x 3 + 1 ) -modules are direct sums of modules of the form Q [ x ] / ( x + 1 ) 2 and Q [ x ] / ( x 2 - x + 1 ) . 2. By the structure theorem for ﬁnitely generated modules over a PID, M = R n T , where T is a torsion module. Since N is not a torsion module, we see that M 6 = T and hence n 6 = 0. We deduce that M = R X for some R -module X (where X = R n - 1 T ) and the ﬁrst statement is proven.

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ahw3 - Math 5126 Monday February 4 Third Homework Solutions...

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