Coalgebraic coalgebras

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.