I'm reading the book "Finite Commutative Rings and their Applications" by G. Bini and F. Flamini. In page 21, the authors state
"The isomorphism $\mathbb Z_{p^n}/pZ_{p^n}\cong\mathbb Z_p$ justifies the fact that an element $u\in\mathbb Z_{p^n}$ can be uniquely written in the form:
$$u=u_0+u_1p+u_2p^2+\ldots+u_{n-1}p^{n-1},$$ where $u_i\in\mathbb Z_p, \;and\;0\leq i\leq n-1.$" $u\in\mathbb Z_{p^n}$
My question is: Why the isomorphism implies this expression for an element?
Moreover, can we generalize this expression to all local finite commutative principal ideal rings? That is, if $R$ is a finite local principal ideal ring and $N$ is its maximal ideal, then any element $u\in R$ can be uniquely written in the form:
$$u=u_0+u_1\alpha+u_2\alpha^2+\ldots+u_{k-1}\alpha^{k-1},$$ where $u_i\in R/N,$ $k$ is the index of nilpotence of $N$, $\alpha$ is a generator of $N$ $\;and\;0\leq i\leq k-1.$" Is that true?
$Z_{p^n}/pZ_{p^n}=Z_p$ let $u_0,...,u_{p-1}\in Z_{p^n}$ such that $u_i$ is identified of $i$ in $Z_p$ if $x\in Z_{p^n}, x=u_{i_1}+v, v\in pZ_{p^n}$, write $v=pu_{i_2}+w\in p^2Z_{p^n}, x=u_{i_1}+pu_{i_2}+w,...$