Consider the following notion of a 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).

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.

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?