Consider the following notion of a [ring object](https://ncatlab.org/nlab/show/ring+object), which works also for (some) non-Cartesian monoidal categories:

> Let $(\mathcal{C},\otimes,\mathbf{1})$ be a monoidal category having Sweedler $\mathrm{Hom}$s and Sweedler products (for instance, we can take $\mathcal{C}$ to be a locally presentable braided monoidal category; see [arXiv:1509.07632, Theorem 4.1](https://arxiv.org/abs/1509.07632)).
> 
> Taking categories of bicommutative Hopf monoids gives us a category
> $\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C})$, which by assumption
> admits a Sweedler product $\boxtimes$ making it into a monoidal
> category.
>
> Then, define a (**commutative**) **ring object** in $\mathcal{C}$ to be a (commutative) monoid in $(\mathsf{HopfMon}^{\mathrm{bicomm}}(\mathcal{C}),\boxtimes,1)$.

Examples:
 - This notion recovers (commutative) rings when applied to $\mathcal{C}=\mathsf{Sets}$, as $\mathsf{BiHopf}^{\mathsf{bicomm}}(\mathsf{Sets})\cong\mathsf{Ab}$ and the Sweedler product in $\mathsf{Ab}$ is the tensor product of abelian groups.
 - More generally, rings in Cartesian monoidal categories coincide with the usual notion of a ring object.
 - Rings in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$ and $(\mathsf{Rings},\otimes_{\mathbb{Z}},\mathbb{Z})$ are [plethories](https://ncatlab.org/nlab/show/plethory).

Moreover, define a **field** in $\mathcal{C}$ to be a commutative ring $R$ in $\mathcal{C}$ such that every module over $R$ is (isomorphic to a) free (one). One may similarly define **skew fields** by considering general rings and left modules.

As an example, this recovers fields and skew fields for $\mathcal{C}=\mathsf{Sets}$.

**Question.** What are some other examples of these two notions? Have they been studied before?