  Riemann zeta values are generalized to multiple zeta values (MZVs) by use of
nested sums, and MZVs are generalized to regularized multiple zeta values
(RMZVs) by regularization of divergent infinite series. In the present paper,
we prove congruence identities of RMZVs of depth $n$ involving a pair of index
sets; the congruence relation is given by the vector space spanned by MZVs of
depth $n-1$ and products of MZVs. We also obtain a proof of the parity result,
and a congruence sum formula for MZVs.
