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# Coalgebraic Coalgebras

Proceedings of the American Mathematical Society
Vol. 45, No. 1 (Jul., 1974), pp. 11-18
DOI: 10.2307/2040598
Stable URL: http://www.jstor.org/stable/2040598
Page Count: 8
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## Abstract

We investigate coalgebras C over a field k such that the dual algebra C* is an algebraic algebra (C is called coalgebraic). The study reduces to the cosemisimple and connected cases. If C is cosemisimple and coalgebraic, then C* 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 an = 0 all $a \in \operatorname{Rad} C^\ast$. By the Nagata-Higman theorem, $\operatorname{Rad} C^\ast$ is nilpotent if nil is characteristic 0.

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