I am trying to understand an ambiguous statement (to me at least).
Let $k=\mathbb{F}_p$ be the field with $p$ elements. Compare the two rings $k[T]/(T^n)$ and $Z/(p^n)$ for $n\geq 2$. Show that their elements can be written uniquely in the form $a_0+a_1T+\cdots+a_{n-1}T^{n-1}$ or $a_0+a_1p+\cdots+a_{n-1}p^{n-1}$ with $a_i \in \{0,1,\cdots,p-1\}$ for $i=0,\cdots,n-1$; determine the addition and multiplication of these power series in the two rings, an note that they differ only by the $p$-adic ''carry''.
I have managed to show that every element can be written in the given forms (the first one is direct and for the second one I first wrote everything in the base $p$, then the terms with powers greater than $n-1$ were included in the ideal). Then, I defined addition and multiplication in the most obvious way (though I am not sure if it is correct, basically I used the base logic and increased the next one by $1$ if the sum exceeds $p-1$). However, my biggest problem is that I do not know what the author means by saying ''p-adic carry''. It would be great if you could check whether I am on the right track and also the ''p-adic carry'' problem of course (my guess is that he means the isomorphism between).
In $k[T]/(T^n)$, if you add $a_0 + a_1 T + \dots + a_{n-1} T^{n-1}$ and $b_0 + b_1 T + \dots + b_{n-1} T^{n-1}$ you get $(a_0 + b_0) + (a_1 + b_1)T + \dots + (a_{n-1} + b_{n-1})T^{n-1}$.
In $Z/(p^n)$, if you add $a_0 + a_1 p + \dots + a_{n-1} p^{n-1}$ and $b_0 + b_1 p + \dots + b_{n-1} p^{n-1}$ you get $(a_0 + b_0) + (a_1 + b_1)p + \dots + (a_{n-1} + b_{n-1})p^{n-1}$. Here, however, $a_i + b_i$ need not be in $\{0, \dots, p-1\}$. To 'repair' that, you change $a_i + b_i$ to $a_i + b_i - p$ and $a_{i+1} + b_{i+1}$ to $a_{i+1} + b_{i+1} - 1$, 'carrying' the $p$, just as you describe. This is very likely what the author means by the '$p$-adic carry'.