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Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in Z/mZ 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 Z/mZ. We relate finite sublattices over Z/kZ for all integers k to an infinite lattice that is embedded in the divisibility lattice on N and to each other as sublattices of this infinite lattice. Using this relation, we generalize several identities on idempotents in Z/mZ 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 Z/mZ.

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It is well-known that for each fixed n and e, the number of subgroups of index p e in Z n is a polynomial in p. Is this true for subrings in Z n of index p e ? Let f n (k) denote the number of subrings of index k in Z n . We can define the subring zeta function overIs this zeta function uniform? These two questions are closely related.In this paper, we describe what is known about these questions, and we make progress toward answering them in a couple ways. First, we describe the connection between counting subrings of index p e in Z n and counting the solutions to a corresponding set of equations modulo various powers of p. We then show that the number of solutions to certain subsets of these equations is a polynomial in p for any fixed n. On the other hand, we give an example for which the number of solutions to a certain subset of equations is not polynomial. Finally, we give an explicit polynomial formula for the number of 'irreducible' subrings of index p n+2 in Z n .