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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Coalgebraic coalgebras

Author: D. E. Radford
Journal: Proc. Amer. Math. Soc. 45 (1974), 11-18
MSC: Primary 16A24
MathSciNet review: 0357474
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Abstract: We investigate coalgebras $ C$ over a field $ k$ such that the dual algebra $ {C^ \ast }$ is an algebraic algebra ($ C$ is called coalgebraic). The study reduces to the cosemisimple and connected cases. If $ C $ is co-semisimple and coalgebraic, then $ {C^ \ast }$ is of bounded degree. If $ C$ is connected, then $ C$ is coalgebraic if, and only if, every coideal is the intersection of cofinite coideals. Our main result is that if $ C$ is a coalgebra over an infinite field $ k$ and the Jacobson radical $ \operatorname{Rad} {C^ \ast }$ is nil, there is an $ n$ such that $ {a^n} = 0$ all $ a \in \operatorname{Rad} {C^ \ast }$. By the Nagata-Higman theorem, $ \operatorname{Rad} {C^ \ast }$ is nilpotent if nil in characteristic 0.

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Keywords: Algebraic algebra, Jacobson radical, dual algebra of a coalgebra
Article copyright: © Copyright 1974 American Mathematical Society