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