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 $(R, +, \cdot\,)$ is an (associative unital) ring, where $+$ is the operation of element-wise addition on $\mathfrak F_{00}(\mathbb N, R)$; 
 - $(\mathbf{1}_1 + a) \cdot b := \sigma(b) \mathbf{1}_1 + ab + \delta(b)$, where $\mathbf{1}_1$ is the indicator function of $\{1\}$ as a subset of $\mathbb N$.

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).