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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Automorphic-differential identities
and actions of pointed coalgebras on rings


Author: Tadashi Yanai
Journal: Proc. Amer. Math. Soc. 126 (1998), 2221-2228
MSC (1991): Primary 16W20, 16W25, 16W30
MathSciNet review: 1459157
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Abstract: 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 $f(R)=0$.


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Additional Information

Tadashi Yanai
Affiliation: Department of Mathematics, Niihama National College of Technology, 7-1 Yagumo-cho, Niihama, Ehime, 792, Japan
Email: yanai@sci.niihama-nct.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04479-7
PII: S 0002-9939(98)04479-7
Received by editor(s): May 31, 1996
Received by editor(s) in revised form: October 24, 1996, and January 24, 1997
Communicated by: Ken Goodearl
Article copyright: © Copyright 1998 American Mathematical Society