I have considered about Lehmer's conjecture that φ(n)|(n-1) if and only if n is a prime. I generalized the conjecture onto genernal finite commutative ring. The idea is below.

 Set A is a finite commutative ring, and B is its unit group. Then there is a partition of A by B in multipication. Choose the representatives {a1,a2,...ak}, then define a subgroup Bi of B as Bi={b is belong to B, b*ai=ai}, for 1=<i<=k. Clearly, sum<1 to k>[B:Bi]=|A|-1, for {ai} is the collection of representatives. With a little more consideration, this converts to the well-known result in basic number theory that sum<d|n>φ(d)=n (using Chinese residue theorem, or fact that every artin ring is a direct sum of local rings). Now, what I conjectured is, if |B| can divide |A|, then A must be a field.
 I don't know much about finite commutative rings, but I can see its significance in number theory here. Moreover, I got to know some analytic methods used in the attacking of Lehmer's conjecture, such as Kevin Ford from UIUC. When faced the problem about rings, some representation theory may work. I don't know.