Let $(n\_1,\ldots, n\_i,\ldots)$ be an infinite tuple of nonnegative integers. Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I\_1,\dots, I\_n , \ldots$ its nonzero ideals (by assumption $D/I\_i$ are finite) such that #$D/I\_i=n\_i$ for each $i$?



Abstract number ring of any characteristic
I'm not sure what you mean by an abstract number ring; nevertheless, here's an attermpt at an answer. If $D$ is to have characteristic $p$, then each of the quotients $D/I\_i$ will be a vector space over $\mathbb Z/p$, so each $n\_i$ will have to be a power of $p$. Conversely, if all the $n\_i$ are powers of $p$, say $n\_i=p^{k\_i}$, then you could take $D$ to be the polynomial ring $(\mathbb Z/p)[X]$ and take $I\_i$ to be the ideal generated by $X^{k\_i}$.


