Came across this question while reviewing for an exam and I'm pretty stumped. We let be a Principal Ideal Domain (P.I.D.) and be a free module of rank 2 with some basis set . Then define to be the submodule of generated by (this is also a free module since is a P.I.D.). Let for some . For , show that is a direct summand of (i.e there exists some submodule of such that ) iff is invertible in . Furthermore for any , is a direct summand of iff the ideal . If , , and , find some such that . Any help here would be appreciated thanks.