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This is the first one in a series of two papers on the continuation of ourstudy in cup products in Hopf cyclic cohomology. In this note we constructcyclic cocycles of algebras out of Hopf cyclic cocycles of algebras andcoalgebras. In the next paper we consider producing Hopf cyclic cocycle from"equivariant" Hopf cyclic cocycles. Our approach in both situations is based on(co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilbertheorem which is different from the old definition of cup products defined viatraces and cotraces on DG algebras and coalgebras.

Cup products in Hopf cyclic cohomology via cyclic modules I