# Name of some commutative ring akin to $p$-adics

I need help in identifying the naming convention of some commutative ring described below.

Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list of positive integers. For each $u\ (0\leq u \leq k)$, define $$E_{u} = \sum_{0 \leq w < u} e_w$$ to be the $u$-th prefix sum of $\textbf{e}$. In my work, I have encountered the set of finite series of the following form: $$\sum_{0\leq u < k} \alpha_u p^{E_u}$$ where $$\alpha_u \in \{0,\ldots, p^{e_u}-1\}.$$

Without distracting the reader with the details of addition and multiplication, I was hoping that such a set looks familiar to the audience. For example, when $e_u = 1$ for all $u$, then we have (up to isomorphism) the familiar representation of the commutative ring $\mathbb{Z}/p^k\mathbb{Z}$.

I have searched the literature for some detail/background on the naming convention of such representations of $\mathbb{Z}/p^{E_k}\mathbb{Z}$--without success. Any ideas where to look?

• Every integer between $0$ and $p^{E_k}-1$ will have a unique representation as a series of this form. Is the ring you have in mind just $\mathbb{Z}/p^{E_k}\mathbb{Z}$? – Julian Rosen Oct 30 '15 at 20:08
• Yes; however, the representation of the integer is of particular interest to me. – Andrey Rukhin Oct 30 '15 at 20:10
• Allow me to clarify: I am looking for the naming convention of such representations of $$\mathbb{Z}/p^{E_k}\mathbb{Z}$$ (if such a thing exists) – Andrey Rukhin Oct 30 '15 at 20:51
• This is an example of a mixed radix numeral system – Julian Rosen Oct 30 '15 at 23:30