codes, write $R:=\F_q[x;\theta]$ and consider the following one-to-one correspondence: $$\label{cod:pi3}
\begin{aligned}
\pi \colon \phantom{aaaaat}\F^n_q\phantom{aaaaa} &\longrightarrow \phantom{aaa} R/R(x^n-\alpha)\\
    (a_0,a_1,\dots,a_{n-1})&\longmapsto a_0+a_1x+\cdots a_{n-1}x^{n-1}\quad .\\
\end{aligned}$$ Note that $\pi$ is an $\F_q$-linear isomorphism of vectorial spaces over $\F_q$. So, we can identify $\F^n_q$ with $R/R(x^n-\alpha)$ and any vector $\vec{a}=(a_0,a_1,\dots,a_{n-1})\in \F^n_q$ with the polynomial class $\pi(\vec{a}):=\sum_{i=0}^{n-1}a_ix^i\in R/R(x^n-\alpha)$.

Put $m:=|\langle \theta \rangle|$. If either $m\nmid n$ or $\alpha\notin\F_q^\theta$, we know that $R/R(x^n-\alpha)$ is not a ring and we cannot argue about its ideals, as in the commutative case. For instance, when $\alpha =1$ the condition $m\mid n$ is assumed (e.g., see [@B0] and [@B2]) and one can construct a one-to-one correspondence between skew cyclic codes and the ideals of $R/R(x^n-1)$. On the other hand, the set $R/R(x^n-\alpha)$ could be considered a left $\F_q$-module or a left $\F_q[x;\theta]$-module.

The next two results give an equivalent definition of skew constacyclic codes and some of their well-known properties.

A nonempty subset $\mathscr{C}\subset\F^n_q$ is a skew $\alpha$-cyclic code if and only if $\pi(\mathscr{C})$ is a left $R$-submodule of the left $R$-module $R/R(x^n-\alpha)$.

\[GMaruta:1\] Let $\pi(\mathscr{C})$ be a left $R$-submodule of $R/R(x^n-\alpha)$ with $R:=\F_q[x;\theta]$, i.e. $\mathscr{C}$ is a skew $\alpha$-cyclic code of $\F_q^n$. Then there exists a unique monic polynomial of 