Consider the base 10 number $124273233$. This can obviously represented as $\{1,2,4,2,7,3,2,3,3\}$ where all characters are representatives of residue classes in $\mathbb Z / 10 \mathbb Z$. The behavior of each register (considered on its own) is as $\mathbb Z / 10 \mathbb Z$, so, to some extent, I can think of each register as a copy of $\mathbb Z / 10 \mathbb Z$.
Let's generalize this: Let $\mathbf C$ be a finite commutative ring with identity (for simplicity, let it be some $\mathbb Z /n \mathbb Z$ with $n$ an arbitrary integer, but I'm actually interested in the case where $\mathbf C$ is a product of $J$ rings of integers modulo $n_j$.), and define $\mathbf C'=\oplus_{i=0}^\infty C$.
Define $\mathbf C''$ equivalent to $\mathbf C'$ as a set, but modify the addition as follows: Let $\boxplus: \mathbf C'' \times \mathbf C''\rightarrow \mathbf C'': \forall i$, (where $i$ is as above), $$ \{\ldots, a_{i+1}, b_i, \ldots\} \boxplus \{\ldots, c_{i+1}, d_i, \ldots\} = \{\ldots, \left(a_{i+1} + c_{i+1} + \left((b_i + d_i) // n\right)\right)\mod n, (b_i + d_i)\mod n, \ldots \}$$ where $ q // r $ is integer division, i.e. I want addition to work the way that it does for registers in base 10, 2, 8, etc. Define multiplication similarly.
Question: What is this algebraic structure called? What should I be looking for when I am trying to understand this structure? It seems like this should be a commutative ring and that, if I'm doing this for arbitrary products of $\mathbb Z / n_j \mathbb Z$ I should recover $\mathbb N \times^{\vert J\vert } \mathbb N$, but I'm not sure I've got the structure right. Could anyone clarify this situation for me?