1. The problem statement, all variables and given/known data Find all ring homomorphisms from 3Z to Z, where 3Z are the integers that are of multiple 3. 2. Relevant equations 3. The attempt at a solution So 3Z is cyclic so σ(3) is sufficient to look. Now all of the other examples have finite groups, so |σ(a)| divides the |a| in the domain. Or, the homomorphisms is define with Z as the domain, and then a=a^2. I understand these examples, but then dont know how to start this one... <3> generates the group, so I know I want to consider it. But if σ(3)=a, but then what? Other examples are a=σ(1)=σ(1*1)=a^2, but that property doesn't hold here, and 1 is not in3Z. I know I'm missing something simple, just can't quite figure out what property I need to put my finger on to make that first step.. Thanks for any help!!