Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let $R$ be a ring, $R _{\mathcal{F}}$ its left Martindale quotient ring and $\mathfrak{A}$ a right ideal of $R$ having no nonzero left annihilator. (1) Let $C$ be a pointed coalgebra which measures $R$ such that the group-like elements of $C$ act as automorphisms of $R$. If $R$ is prime and $\xi \cdot \mathfrak{A}=0$ for $\xi \in R\#C$, then $\xi \cdot R=0$. Furthermore, if the action of $C$ extends to $R _{\mathcal{F}}$ and if $\xi \in R _{\mathcal{F}}\#C$ such that $\xi \cdot \mathfrak{A}=0$, then $\xi \cdot R _{\mathcal{F}}=0$. (2) Let $f$ be an endomorphism of $R _{\mathcal{F}}$ given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If $R$ is semiprime and $f(\mathfrak{A})=0$, then $f(R)=0$.