Coalgebraic coalgebras

Author:
D. E. Radford

Journal:
Proc. Amer. Math. Soc. **45** (1974), 11-18

MSC:
Primary 16A24

DOI:
https://doi.org/10.1090/S0002-9939-1974-0357474-9

MathSciNet review:
0357474

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Abstract: We investigate coalgebras over a field such that the dual algebra is an algebraic algebra ( is called *coalgebraic*). The study reduces to the cosemisimple and connected cases. If is co-semisimple and coalgebraic, then is of bounded degree. If is connected, then is coalgebraic if, and only if, every coideal is the intersection of cofinite coideals. Our main result is that if is a coalgebra over an infinite field and the Jacobson radical is nil, there is an such that all . By the Nagata-Higman theorem, is nilpotent if nil in characteristic 0.

**[1]**I. N. Herstein,*Theory of rings*, Math. Lecture Note Series, University of Chicago, Chicago, Ill., 1961.**[2]**Robert G. Heyneman and David E. Radford,*Reflexivity and coalgebras of finite type*, J. Algebra**28**(1974), 215–246. MR**0346001**, https://doi.org/10.1016/0021-8693(74)90035-0**[3]**Robert G. Heyneman and Moss Eisenberg Sweedler,*Affine Hopf algebras. I*, J. Algebra**13**(1969), 192–241. MR**0245570**, https://doi.org/10.1016/0021-8693(69)90071-4**[4]**David E. Radford,*Coreflexive coalgebras*, J. Algebra**26**(1973), 512–535. MR**0327818**, https://doi.org/10.1016/0021-8693(73)90012-4**[5]**Moss E. Sweedler,*Hopf algebras*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR**0252485**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0357474-9

Keywords:
Algebraic algebra,
Jacobson radical,
dual algebra of a coalgebra

Article copyright:
© Copyright 1974
American Mathematical Society