Assess students' ability to compute Ext^1_R(M,N) and Tor_1^R(M,N) for finitely generated modules over a PID R by using the structure theorem to reduce to cyclic summands, determine the resulting abelian groups, interpret Ext^1 as equivalence classes of extensions (split vs non‑split), and classify all extension types in concrete examples (e.g., R = Z with modules like Z, Z/nZ, direct sums). Expect explicit calculations, justification via resolution or Hom/Tensor adjunctions, and clear classification statements.