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Automorphic-differential identities and actions of pointed coalgebras on rings

Authors:	Tadashi Yanai

Institution:	Department of Mathematics, Niihama National College of Technology, 7-1 Yagumo-cho, Niihama, Ehime, 792, Japan

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

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