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On radicals of module coalgebras
Authors:Yuqun Chen  Kar-Ping Shum
Institution:a School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China
b Department of Mathematics, Iowa State University, Ames, IA 50011, USA
c Faculty of Science, The Chinese University of Hong Kong, Hong Kong, China
Abstract:We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming a bijective antipode.
Keywords:16W30
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