algebraic structure of modular multiplication

Question

Consider the set ${\mathbb Z}_N = \{0, \ldots, N-1\}$ under multiplication modulo $N$. When $N=pq$ with $p, q$ relatively prime, ${\mathbb Z}_N$ and ${\mathbb Z}_p \times {\mathbb Z}_q$ are isomorphic as monoids. Is there some similarly ''nice'' characterization of the monoid ${\mathbb Z}_{p^k}$? (I am aware of such a characterization for the group of invertible elements modulo $p^k$, but I specifically looking for the monoid including the non-invertible elements.)

Answer 1

I don't know what kind of characterisation you're looking for, but it can't split as a product:

For $\mathbb{Z}_{p^k}$ to split as a product $M \times N$, the factor monoids must have orders $p^m$ and $p^n$. And an element of $\mathbb{Z}_{p^k}$ is nilpotent iff both its components are nilpotent in the factor monoids. So the number of nilpotent elements, $p^{k-1}$ is the product of the number of nilpotent elements in the factor monoids, $n_M$ and $n_N$. But $p^{k-1}$ only factors as powers of $p$, and there isn't any way of choosing powers of $p$, $p^m$, $p^n$, $n_M$ and $n_N$, such that $n_M < p^m$ and $n_N < p^n$ and $n_M n_N = p^{m+n-1}$.

Answer 2

When $p$ is odd, or $p=2$ and $p^k\in \{2,4\}$, then there exists a primitive root modulo $p^k$. Therefore, the monoid $\mathbb{Z}/p^k\mathbb{Z}$ under $\times$ is a disjoint union of the group $(\mathbb{Z}/p^k\mathbb{Z})^\times$ (isomorphic to $\mathbb{Z}/\phi(p^k)\mathbb{Z}=\mathbb{Z}/p^{k-1}(p-1)\mathbb{Z}$ under addition) and the nilsemigroup $p\mathbb{Z}/p^k\mathbb{Z}$.

https://en.wikipedia.org/wiki/Primitive_root_modulo_n

https://en.wikipedia.org/wiki/Nilsemigroup