$\mathbb{Z}/(n+1)\mathbb{Z}\mapsto[n]=\{0,1,...n\}$

Question

Does this mapping between objects $$ \mathbb{Z}/(n+1)\mathbb{Z}\mapsto[n]=\{0,1,...n\} $$ for all n > or = to 0

mapping between morhisms $$ (\mathbb Z/(n+1)\mathbb Z\to\mathbb Z/(m+1)\mathbb Z)\mapsto([n]=\{0,1,...n\}\to[m]=\{0,1,...m\}) $$

induces a forgetful functor, forgetting the congruence relation, between a subcategory of the category of rings (the one in which objects are of the above form) to the simplex category ?

Answer 1

No, for two reasons:

• There are rings that aren't isomorphic to a ring of the form $\mathbb{Z}/(n+1)\mathbb{Z}$, and you haven't specified how the alleged functor should act on those
• You haven't specified what to do on morphisms. There is no explicit indication, and I don't see what you've written as implicitly suggesting any such action.