Singleton bounds for R-additive codes

Shiromoto (Linear Algebra Applic 295 (1999) 191-200) obtained the basic exact sequence for the Lee and Euclidean weights of linear codes over \begin{document}$ \mathbb{Z}_{\ell}$\end{document} and as an application, he found the Singleton bounds for linear codes over \begin{document}$ \mathbb{Z}_{\ell}$\end{document} with respect to Lee and Euclidean weights. Huffman (Adv. Math. Commun 7 (3) (2013) 349-378) obtained the Singleton bound for \begin{document}$ \mathbb{F}_{q}$\end{document} -linear \begin{document}$ \mathbb{F}_{q^{t}}$\end{document} -codes with respect to Hamming weight. Recently the theory of \begin{document}$ \mathbb{F}_{q}$\end{document} -linear \begin{document}$ \mathbb{F}_{q^{t}}$\end{document} -codes were generalized to \begin{document}$ R$\end{document} -additive codes over \begin{document}$ R$\end{document} -algebras by Samei and Mahmoudi. In this paper, we generalize Shiromoto's results for linear codes over \begin{document}$ \mathbb{Z}_{\ell}$\end{document} to \begin{document}$ R$\end{document} -additive codes. As an application, when \begin{document}$ R$\end{document} is a chain ring, we obtain the Singleton bounds for \begin{document}$ R$\end{document} -additive codes over free \begin{document}$ R$\end{document} -algebras. Among other results, the Singleton bounds for additive codes over Galois rings are given.