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B & C, please show all working

4. (a) State the structure theorem for finitely generated modules over PIDs. [3 marks) (b) Prove that every square matrix with coefficients in k has a rational canonical form. [10 marks] (c) Let M = Z4 and let L be the submodule of M generated by 2 0 aj 0 2 0 2 0 Compute the rank of M/L and the invariant factors of M/L. [12 marks] END OF PAPER 4. (a) State the structure theorem for finitely generated modules over PIDs. [3 marks) (b) Prove that every square matrix with coefficients in k has a rational canonical form. [10 marks] (c) Let M = Z4 and let L be the submodule of M generated by 2 0 aj 0 2 0 2 0 Compute the rank of M/L and the invariant factors of M/L. [12 marks] END OF PAPER