The mapping $mathbb{Z} to mathbb{Z}_n$ defined by $a mapsto a pmod{n}$ is commonly denoted as such. Or, in the construction or interpretation of $mathbb{Z}_n$ involving the congruence classes modulo $n$ of $mathbb{Z}$, I’ve seen that same mapping denoted $a mapsto (a)$, or $amapsto (a)_n$.

What would be the clearest way to denote a mapping $mathbb{Z}_m to mathbb{Z}_n$ where $n leq m$ and such that $a mapsto a$ where the LHS of that mapping is interpreted as $a in mathbb{Z}_m$ and the RHS is interpreted as $a in mathbb{Z}_n$?

Also, how would I denote the same kind of mapping $mathbb{Z}(t) to mathbb{Z}_n(t)$ where I want $p(t) = a_0 + a_1t + cdots + a_kt^k$ to map the polynomial $p(t) = b_0 + b_1t+cdots + b_kt^k$ where $b_i = a_i pmod{n}$ for each $0 leq i leq k$? i.e. that I want the same polynomial $p(t)$ but with its coefficients interpreted in $mathbb{Z}_n$ as opposed to in $mathbb{Z}$.