When is the unit group of a ring R cyclic?

Question

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?"

score 1 · Answer 1 · answered Apr 02 '18 at 08:34

1

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

answered Apr 02 '18 at 08:34

Dietrich Burde

130,978

That answer only settles things for commutative rings. What about non-commutative rings? – Keshav Srinivasan Apr 02 '18 at 08:48
In general, there is no such answer possible. Even the question, what other groups besides cyclic ones can occur, is too difficult - see here. – Dietrich Burde Apr 02 '18 at 08:58

When is the unit group of a ring R cyclic?

1 Answers1