For some rings, the unit group is a cyclic group. For other rings, that's not the case. My question is, is there some method of determining whether the unit group of a ring is cyclic or not, short of actually examining the unit group?
Like is there a theorem of the form "If a ring has property P, then its unit group is cyclic?"
For finite rings the question has been answered here:
Is the group of units of a finite ring cyclic?
The result is a list of six properties $A,B,C,D,E,F$. On the other hand, if $R$ is a field with cyclic unit group, then $R$ must be already finite:
the unit group of an infinite field cannot be cyclic