  Idempotent elements are a well-studied part of ring theory, with several
identities of the idempotents in $\mathbb{Z}/m\mathbb{Z}$ already known.
Although the idempotents are not closed under addition, there are still
interesting additive identities that can be derived and used. In this paper, we
give several new identities on idempotents in $\mathbb{Z}/ m\mathbb{Z}$. We
relate finite sublattices over $\mathbb{Z}/ k\mathbb{Z}$ for all integers $k$
to an infinite lattice that is embedded in the divisibility lattice on
$\mathbb{N}$ and to each other as sublattices of this infinite lattice. Using
this relation, we generalize several identities on idempotents in
$\mathbb{Z}/m\mathbb{Z}$ to those involving idempotents related to these finite
sublattices. Finally, as an application of the above idempotent identities, we
derive an algorithm for calculating modular exponentiation over $\mathbb{Z}/
m\mathbb{Z}$.
