Let ${\mathfrak{R}}_l={\mathbb{F}}_q[u]/⟨u^l-u⟩$, where $q=p^m$, $p$ is a prime, and $m\in {\mathbf{N}}^{\mathbf{+}}$, $(l-1)\mid (p-1)$. First, we study isometry and equivalence between constacyclic codes over ${\mathfrak{R}}_l$, and we present the equivalence conditions for $(e_1{a}_1+e_2{a}_2+\cdots +e_l{a}_l)$-constacyclic code and $(e_1{b}_1+e_2{b}_2+\cdots +e_l{b}_l)$-constacyclic code. Further, based on the equivalence conditions, we classify constacyclic codes over ${\mathfrak{R}}_l$. Finally, based on the equivalence of constacyclic codes over ${\mathfrak{R}}_l$ and CSS construction, we construct some new quantum codes that are better than the existing codes in some recent references.