  We consider modifications of Manin symbols in first homology groups of
modular curves with p-adic integer coefficients for an odd prime p. We show
that these symbols generate homology in primitive eigenspaces for the action of
diamond operators, with a certain condition on the eigenspace that can be
removed on Eisenstein parts. We apply this to prove the integrality of maps
taking compatible systems of Manin symbols to compatible systems of zeta
elements. In the work of the first two authors on an Iwasawa-theoretic
conjecture of the third author, these maps are constructed with certain bounded
denominators. As a consequence, their main result on the conjecture was proven
after inverting p, and the results of this paper allow one to remove this
condition.
