Let $K$ be a local number field, $L$ a quadratic reduced $K$-algebra over $K$, and let $a\in O_L$ and consider $R = O_K[a]$, where $O_K, O_L$ are the ring of integers of $K, L$, respectively. Under these assumptions we have either $L$ is a field or $L = K\times K$.
When $L$ is a field we have $R\subset O_L$ and hence is an integral domain itself. Hence, because it is monogenic I can deduce it is Gorenstein. This uses Proposition 3.6 of Gorenstein Rings by Jensen and Thorup. In this proposition, they seem to require (according to their general assumptions) that $R$ is already known to be a local integral domain.
My question is what happens in the other case? Is $O_K[a]$ still Gorenstein when $L = K\times K$? The question makes sense but we dont have locality anymore, since $O_L = O_K\times O_K$, and of course is not integral domain. I don't seem to find results that tell me it is, but also nothing that tell me it is not.
Commutative Algebra is not my forte so I do get confused with these things. Any help is much appreciated.