Let $A$ be an integer with its representation in base $p$ ($p$ may be prime number but not necessarily) described as: $$A=(a_ma_{m-1}\ldots a_1a_0)_{p}$$ We know $A\equiv (a_n\ldots a_1a_0)_{p}\pmod {p^n} $ , $m\geq n$.
I'm interested in defining a new semi-modulo operation as follows: $$ A\equiv (a_ma_{m-1}\ldots a_{m-n})_{p} \pmod {p^n}' , m\geq n $$
We also know that the algebraic equation associated with the recurrence $$X_{n+k}+ a_1X_{n+k-1}+a_2X_{n+k-2}+\cdots+a_{k}X_{n}=a$$ is
$$F(x)=x^k+a_1x^{k-1}+\cdots+a_k=0$$
Can this equation be decomposed into prime functions $\phi_i(x)\pmod p'$?
$$F(x)\equiv\phi_1(x)^{e_1}\phi_2(x)^{e_2}\ldots\phi_r(x)^{e_r} \pmod {p}'$$
Does it have the same properties as modulo $p$ operation, for example in studying periodicity of Linear Recurring Sequences $\pmod {p^n}'$?
And how can I apply theorem of Dedekind to it?
Hints:
I choose (mod)' as a symbol of the new operation. I couldn't figure out whether this operation is defined in number theory before.
I think defining such operation may be useful in analyzing binary arithmetic (such as multiplication) of fractional numbers in which $p=2$.