# abstract algebra – Best way to denote mappings \$mathbb{Z}_m to mathbb{Z}_n\$, and \$mathbb{Z}[t] to mathbb{Z}_m[t]\$?

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 $$mathbb{Z}$$.