How does one understand $f:\mathbb{Z}[[t]]\otimes_\mathbb{Z}\mathbb{Z}/I\to\mathbb{Z}/I[[t]]$? I am trying to interpret an exam question. Consider the ring of integers $\mathbb{Z}$ and some ideal $I\subset\mathbb{Z}$. I'm having trouble understanding how a specific mapping is actually defined. Here is the setting: Let $i:\mathbb{Z}/I\to \mathbb{Z}/I[[t]]$ be the inclusion, where $\mathbb{Z}/I[[t]]$ is the ring of formal power series with coefficients in $\mathbb{Z}/I$. Further let $j:\mathbb{Z}[[t]]\to\mathbb{Z}/I[[t]]$ where $j$ reduces the coefficients modulo $I$. And here is what I'm trying to interpret:

Let $f:\mathbb{Z}[[t]]\otimes_\mathbb{Z}\mathbb{Z}/I\to\mathbb{Z}/I[[t]]$ be the map corresponding to $i$ and $j$.

Would one understand the mapping $f$ to do the following:
$$(\sum_ia_it^{i}\otimes \overline{n})\stackrel{f}{\longmapsto}j\left(\sum_ia_it^{i}\right)\cdot i(\overline{n})$$
My thoughts: In this case, $f$ takes elements in the tensor product and actually hits something in the desired codomain, but I'm not sure if I have interpreted it correctly. If I'm wrong, is there an obvious way to interpret this construction? I feel like the exam-question is formulated in a way that suggests that there is 
 A: Yes, the formula you give is almost certainly what they intended. If you read closely you can see that the link provided by KLStream's comment doesn't quite apply, as that's for maps into a tensor product, and you've got two maps into a single ring. But (and this is just my opinion) the tensor product is a concept that is used in almost too many places and people aren't always explicit about which context they are using it. The basic motivation is always "do the right thing on basis elements, and extended linearly", which was what your and KLStream's formulas do. 
This approach skips past the question of whether such a map exists and is uniquely defined. There is usually a universal property (or definition) of a tensor product that answers "yes" to the existence/uniqueness question. It depends on what approach your teacher has taken to tensor products whether or not you need to worry about it. 
As to whether or not there is some other interpretation, it would be interesting to see more of the question, in particular which part makes you suspect that there may be.
