Prime-Power Multiplicative Modular Monoid Morphisms

We undertake a systematic study of multiplicative monoid morphisms between prime-power modular monoids $(\mathbb{Z}\mathrm{/}{p}^e\mathbb{Z},\cdot )$ and $(\mathbb{Z}\mathrm{/}{q}^f\mathbb{Z},\cdot )$, for arbitrary primes $p,q$ and integers $e,f\ge 1$. Our approach proceeds in three stages. We first determine the hom-set $\mathbf{H}\mathbf{o}\mathbf{m}(U_{p^e},U_{q^f})$ of group morphisms between the corresponding unit groups and exhibit explicit isomorphisms describing its group structure. We then identify the subset ${\mathcal{H}}_{p^e,q^f,v}$ of unit group morphisms that extend to full monoid morphisms on $(\mathbb{Z}\mathrm{/}{p}^e\mathbb{Z},\cdot )$, characterizing it via explicit divisibility conditions and describing its algebraic structure. Finally, we give a complete description of ${\mathbf{H}\mathbf{o}\mathbf{m}}_{\mathbf{M}\mathbf{o}\mathbf{n}}((\mathbb{Z}\mathrm{/}{p}^e\mathbb{Z},\cdot ),(\mathbb{Z}\mathrm{/}{q}^f\mathbb{Z},\cdot ))$ and establish an explicit monoid isomorphism between this hom-set and a naturally defined monoid built from unit groups of lower-level modular rings.