  Let $f(u)$ and $g(v)$ be two polynomials of degree $k$ and $\ell$
respectively, not both linear, which split into distinct linear factors over
$\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle
f(u),g(v),\\uv-vu\rangle$ be a finite commutative non-chain ring. In this
paper, we study $\psi$-skew cyclic and $\theta_t$-skew constacyclic codes over
the ring $\mathcal{R}$ where $\psi$ and $\theta_t$ are two automorphisms
defined on $\mathcal{R}$.
