  We introduce non-associative Ore extensions, $S = R[X ; \sigma , \delta]$,
for any non-associative unital ring $R$ and any additive maps $\sigma,\delta :
R \rightarrow R$ satisfying $\sigma(1)=1$ and $\delta(1)=0$. In the special
case when $\delta$ is either left or right $R_{\delta}$-linear, where
$R_{\delta} = \ker(\delta)$, and $R$ is $\delta$-simple, i.e. $\{ 0 \}$ and $R$
are the only $\delta$-invariant ideals of $R$, we determine the ideal structure
of the non-associative differential polynomial ring $D = R[X ; \mathrm{id}_R ,
\delta]$. Namely, in that case, we show that all ideals of $D$ are generated by
monic polynomials in the center $Z(D)$ of $D$. We also show that $Z(D) =
R_{\delta}[p]$ for a monic $p \in R_{\delta}[X]$, unique up to addition of
elements from $Z(R)_{\delta}$. Thereby, we generalize classical results by
Amitsur on differential polynomial rings defined by derivations on associative
and simple rings. Furthermore, we use the ideal structure of $D$ to show that
$D$ is simple if and only if $R$ is $\delta$-simple and $Z(D)$ equals the field
$R_{\delta} \cap Z(R)$. This provides us with a non-associative generalization
of a result by \"{O}inert, Richter, and Silvestrov. This result is in turn used
to show a non-associative version of a classical result by Jordan concerning
simplicity of $D$ in the cases when the characteristic of the field $R_{\delta}
\cap Z(R)$ is either zero or a prime. We use our findings to show simplicity
results for both non-associative versions of Weyl algebras and non-associative
differential polynomial rings defined by monoid/group actions on compact
Hausdorff spaces.
