  Given a non-associative unital ring $R$, a monoid $G$ and a set $\pi$ of
additive maps $R \rightarrow R$, we introduce the Ore monoid ring $R[\pi ; G]$,
and, in a special case, the differential monoid ring. We show that these
structures generalize, in a natural way, not only the classical Ore extensions
and differential polynomial rings, but also the constructions, introduced by
Cojuhari, defined by so-called $D$-structures $\pi$. Moreover, for commutative
monoids, we give necessary and sufficient conditions for differential monoid
rings to be simple. We use this in a special case to obtain new and shorter
proofs of classical simplicity results for differential polynomial rings in
several variables previously obtained by Voskoglou and Malm by other means. We
also give examples of new Ore-like structures defined by finite commutative
monoids.
