Wu and Shi [‘A note on k-Galois LCD codes over the ring \mathbb {F}_q + u\mathbb {F}_q$\mathbb {F}_q + u\mathbb {F}_q$ ’, Bull. Aust. Math. Soc. 104(1) (2021), 154–161] studied k $k$ -Galois LCD codes over the finite chain ring \mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q$\mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q$ , where u^2=0$u^2=0$ and q=p^e$q=p^e$ for some prime p and positive integer e. We extend the results to the finite nonchain ring \mathcal {R} =\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q$\mathcal {R} =\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q$ , where u^2=u,v^2=v $u^2=u,v^2=v$ and uv=vu $uv=vu$ . We define a correspondence between the l $l$ -Galois dual of linear codes over \mathcal {R} $\mathcal {R}$ and the l $l$ -Galois dual of their component codes over \mathbb {F}_q $\mathbb {F}_q$ . Further, we construct Euclidean LCD and l $l$ -Galois LCD codes from linear codes over \mathcal {R} $\mathcal {R}$ . We prove that any linear code over \mathcal {R} $\mathcal {R}$ is equivalent to a Euclidean code over \mathbb {F}_q$\mathbb {F}_q$ with q>3 $q>3$ and an l $l$ -Galois LCD code over \mathcal {R}$\mathcal {R}$ with 0<l<e$0<l<e$ and p^{e-l}+1\mid p^e-1$p^{e-l}+1\mid p^e-1$ . Finally, we investigate MDS codes over \mathcal {R}$\mathcal {R}$ .