Adv Alegbra HW Solutions 110

Adv Alegbra HW Solutions 110 - 110 some constant c. If x is...

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110 some constant c .If x is a multiple of ax + by + c , then b = 0; if y is a mul- tiple, then a = 0. We conclude that d ( x , y ) is a nonzero constant. Since k is a f eld, d is a unit, and so ( x , y ) = ( d ) = k [ x , y ] , a contradiction. 3.76 For every m 1, prove that every ideal in I m is a principal ideal. (If m is composite, then I m is not a PID because it is not a domain.) Solution. Let I be an ideal in I m . Consider the ring homomorphism ν : Z I m de f ned by ν : n 7→[ n ] ; of course, ν is a surjection. It is easily seen that the inverse image J = ν 1 ( I ) ={ n Z : ν( n ) I } is an ideal in Z ; moreover, ν( J ) = I . But every ideal in Z is principal, so that J = ( a ) . It follows that I = ( [ a ] ) . This argument really shows that if R is a PID and L is an ideal in R , then every ideal in the quotient ring R / L is principal. 3.77 Let R be a PID and let π R be an irreducible element. If β R and π - β , prove that π and β
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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