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.