On the structure of submodule of finitely generated module over PID

TL;DR

This paper investigates the structure of submodules within finitely generated modules over principal ideal domains, highlighting specific counterexamples and errors in existing theorems related to subgroup containment and cyclic generation.

Contribution

It identifies a flaw in a theorem about submodule containment in finitely generated modules over PIDs and clarifies the conditions under which submodules are cyclic.

Findings

Counterexample showing the theorem's failure

Identification of an error in the proof regarding preimages

Clarification of conditions for cyclic submodules

Abstract

Consider $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$, and the subgroup generated by $(2,1)$, it is a subgroup isomorphic to $\mathbb{Z}/4\mathbb{Z}$. If my theorem holds, it must contained in a cyclic group generated by some element of order $8$ which is impossible. The error in 4.2 is because "preimage" do not preserce the image of $[x\mapsto px]$.