In this article, for the finite field \({\mathbb {F}}_q\) , we show that the \({\mathbb {F}}_q\) -algebra \({\mathbb {F}}_q[x]/\langle f(x) \rangle \) is isomorphic to the \({\mathbb {F}}_q\) -algebra \({\mathbb {F}}_q^{\deg f(x)}\) if and only if f(x) splits over \({\mathbb {F}}_q\) into distinct factors. More generally, we show that if \(f_i(x_i)\) splits into distinct factors over \({\mathbb {F}}_q,\) \({\mathbb {F}}_q\) -algebra \({\mathbb {F}}_q[x_1, x_2,\dots , x_k]/\langle f_1(x_1), f_2(x_2),\dots , f_k(x_k)\rangle \) is isomorphic to \({\mathbb {F}}_q^l,\) for some \(l\in {\mathbb {N}}.\) On the other hand, we establish that every finite-dimensional \({\mathbb {F}}_q\) -algebra \({\mathcal {S}}\) has an orthogonal basis of idempotents with their sum equal to \(1_{{\mathcal {S}}}\) if and only if \({\mathcal {S}}\cong {\mathbb {F}}_q^l\) as \({\mathbb {F}}_q\) -algebras, where \(l=\dim _{{\mathbb {F}}_q} {\mathcal {S}}\) . Instead of studying polycyclic codes over \({\mathbb {F}}_q\) -algebras \({\mathbb {F}}_q[x_1, x_2,\dots , x_k]/\langle f_1(x_1), f_2(x_2),\dots , f_k(x_k)\rangle \) where \(f_i(x_i)\) splits into distinct linear factors over \({\mathbb {F}}_q,\) which is a subclass of \({\mathbb {F}}_q^l,\) we study polycyclic codes over \({\mathbb {F}}_q^l\) and obtain their unique decomposition into polycyclic codes over \({\mathbb {F}}_q\) for every such orthogonal basis of \({\mathbb {F}}_q^l\) . We refer to it as an \({\mathbb {F}}_q\) -decomposition. An \({\mathbb {F}}_q\) -decomposition enables us to use results of polycyclic codes over \({\mathbb {F}}_q\) to study polycyclic codes over \({\mathbb {F}}_q^l\) ; for instance, we show that the annihilator dual of a polycyclic code over \({\mathbb {F}}_q^l\) is a polycyclic code over \({\mathbb {F}}_q^l\) . These results extend to finite commutative semisimple rings. Furthermore, with the help of different Gray maps, we produce a good number of examples of MDS or almost-MDS or/and optimal codes; some of them are LCD over \({\mathbb {F}}_q\) . Finally, we study Gray maps from \(({\mathbb {F}}_q^l)^n\) to \({\mathbb {F}}_q^{nl},\) and use them to construct quantum codes with the help of CSS construction.
