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I'm stuck on a couple of practice problems relating to PIDs, they are paraphrased below:

Given a PID $R$ with $a$ and $b$ in $R$ and gcd$(a,b)=1$ I need to show that:

1) There are elements $s$ and $t$ of $R$ such that $sa+tb=1$

2) The $R$-module $R/(a) \bigoplus R/(b)$ is isomorphic to the $R$-module $R/(ab)$ (where $(a)$ denotes the ideal generated by $a$)

I know that for 1 if I can assume that $(a)+(b)=R$ because gcd$(a,b)=1$ then I am done, but I'm not sure I can assume that (i.e. I can't remember if the poof of that fact uses 1).

For 2 I'm not really sure where to start and would appreciate a full explained answer as the test I am practicing for is tomorrow.

  • 0
    For 1), use the fact that $(a)+(b)$ is a principal ideal.2012-12-04
  • 0
    Or give us a definition of $\gcd(a,b)$ (for question 1).2012-12-04

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