Is the group of units of a finite ring cyclic?

Question

The group of units of a finite field is cyclic. Is it true that the group of units of a finite ring is also cyclic? If not, where does the ring structure prevents us from obtaining the result that is true for fields?

Answer 1

A quick google search finds the paper Finite Rings Having a Cyclic Multiplicative Group of Units by Robert W. Gilmer, Jr. which settles the question as follows:

This is complemented by proving that every finite commutative ring is a direct sum of primary rings and that its group of units is the direct product of the group of units of the primary rings, and is cyclic iff each part is cyclic and they have coprime orders.

Answer 2

No, $\mathbb Z/8$ has unit group $\{1,3,5,7\}$ mod 8 which is a 2,2 group.

In number-theoretic situations there are coherent things that can be said, and/but in general I think nothing decisive can be said.

Answer 3

Given any ring $A$ with non-trivial group of units $A^\times$, the ring $A\times A$ will have a non-cyclic group of units.

This is because for rings $A$ and $B$ one has $(A\times B)^\times=A^\times\times B^\times$ and if $G$ is a non-trivial group the group $G\times G$ is never cyclic.