To my knowledge (see, e.g., H.H. Brungs and G. Törner's Skew Power Series Rings and Derivations [J. Algebra 87 (1984), 368-379]), skew polynomial rings were first introduced by Ø. Ore in 1933: Given a ring $R$, a ring endomorphism $\sigma$ of $R$, and a $\sigma$-derivation $\delta$, we can uniquely define a (binary) operation $\cdot$ on the set $\mathfrak F_{00}(\mathbb N, R)$ of all finitely supported functions $\mathbb N \to R$ such that

• the triple $(\mathfrak F_{00}(\mathbb N, R), +, \cdot\,)$ is an (associative unital) ring, where $+$ is the operation of element-wise addition on $\mathfrak F_{00}(\mathbb N, R)$, and
• $(\mathbf{1}_1 + a\mathbf{1}_0) \cdot b\mathbf{1}_0 = \sigma(b) \mathbf{1}_1 + (ab + \delta(b))\mathbf{1}_0$ for all $a, b \in R$, where $\mathbf{1}_i$ is, for each $i \in \mathbb N$, the indicator function of $\{i\}$ as a subset of $\mathbb N$ and we are viewing the group $(\mathfrak F_{00}(\mathbb N, R), +)$ as a left $R$-module in the obvious way.

Now, in the same way as polynomial rings (in arbitrarily many variables) are generalized by monoid rings (so recovering, among many others, Laurent polynomial rings and free algebras in one go), it seems natural to me to extend Ore's definition to skew monoid rings (however defined) and thence to skew ordered series rings (which, however defined, should in turn provide a natural generalization of ordered series rings in the sense of P.M. Cohn's 2006 book on FIRs and localization — AFAIK, these series rings were first defined by R.E. Johnson in Unique factorization monoids and domains [Proc. Amer. Math. Soc. 28 (1971), No. 2, 397-404], but please correct me if I'm wrong in my conclusions). So the question is:

Q. Where can I read about skew monoid rings and skew ordered series rings (giving for granted that they have been considered before in the literature)?

I could only dig up papers dealing with special cases of what I'd consider a satisfactory definition of a skew monoid ring (resp., a skew ordered series ring), and I wonder if there is a good reason for that (beyond the fact that people may have found such a thing not especially interesting after all).