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 an (associative, unital) ring $R$, a ring endomorphism $\sigma$ of $R$, and a $\sigma$-derivation $\delta$, we can uniquely${}^{(1)}$ 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 a ring, where $+$ is the operation of pointwise addition on $\mathfrak F_{00}(\mathbb N, R)$, and
- $(\chi_1 + a\chi_0) \cdot b\chi_0 = \sigma(b) \chi_1 + (ab + \delta(b))\chi_0$ for all $a, b \in R$, where $\chi_i$ 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${}^{(2)}$.
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 then 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${}^{(3)}$). So the question is:
Q. Where can I read about skew monoid rings [edit: answered by Benjamin Steinberg in a comment${}^{(4)}$] 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).
Notes.
(1) See, e.g., Sect. 5.2 in [P.M. Cohn, Introduction to Ring Theory, Springer, 2004 (3rd ed.)].
(2) You may want to introduce a formal variable $X$ and set $X^i := \chi_i$ (see Pace Nielsen's comment).
(3) AFAIK, ordered series rings were first introduced by R.E. Johnson in Unique factorization monoids and domains [Proc. Amer. Math. Soc. 28 (1971), No. 2, 397-404] (please correct me if I'm wrong on this point). One starts with a linearly ordered monoid $\mathcal H = (H, \preceq)$ and consider the set $\mathcal W(\mathcal H, R)$ of all functions $f \colon H \to R$ whose support $s(f) := H \setminus f^{-1}(0_R)$ is well ordered under $\preceq$ (in the sense that the restriction of $\preceq$ to $s(f)$ results in a well-ordered set). If $f, g \in \mathcal W(\mathcal H, R)$, then $$ \Gamma(u) := \{(x, y) \in H \times H \colon xy = u \text{ and } f(x) g(y) \ne 0_R\} $$ is a finite set for every $u \in H$ (see Sect. 2 in Johnson's paper), so one can turn $\mathcal W(\mathcal H, R)$ into a ring $R[\![\mathcal H]\!]$ by taking the sum $f+g$ of $f$ and $g$ to be their pointwise sum and the product $fg$ of $f$ by $g$ to be the function $H \to R \colon u \mapsto \sum_{(x, y) \in \Gamma(u)} f(x) g(y)$. Cohn refers to $R[\![\mathcal H]\!]$ as the ring of ordered series of $\mathcal H$ over $R$.
(4) The reference provided by Benjamin Steinberg is [E.P. Cojuhari, Monoid algebras over non-commutative rings, Internat. Electron. J. Algebra 2 (2007) 28-53], where the author seems to expand on an idea of Brungs and Törner's paper (the same cited at the beginning of this post), but only for skew monoid rings (whereas Brungs and Törner deal with rings of skew power series).
