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We consider non-monogenic simplest cubic fields $K=\mathbb{Q}(\rho)$ in thefamily introduced by Shanks, and among these, we focus in the fields whosegeneralized module index $[\mathcal{O}_K:\mathbb{Z}[\rho]]$ is a prime number$p$. We prove that these fields arise exactly for $p=3$ or$p\equiv1\,(\mathrm{mod}\,6)$ and we use the method introduced inarXiv:2005.12312 to find the additive indecomposables of $\mathcal{O}_K$. Wedetermine the whole structure of indecomposables for the family with $p=3$ andobtain that the behaviour is not uniform with respect to the indecomposables of$\mathbb{Z}[\rho]$. From the knowledge of the indecomposables we derive somearithmetical information on $K$, namely: the smallest and largest norms ofindecomposables, the Pythagoras number of $\mathcal{O}_K$ and bounds for theminimal rank of universal quadratic forms over $K$.

Additive structure of non-monogenic simplest cubic fields