The well known [Dirichlet's unit theorem](https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem) states that the unit group of a maximal order in a quadratic number field is *finitely generated* of rank blah blah blah. I think it's pretty naive to expect a most radical generalization to hold, namely:



> 
> **Generalized Dirichlet unit theorem**. The unit group of an associative unital ring with finitely generated additive group is finitely generated.
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Is there any concrete counterexample?



Example of a ring with non-finitely generated unit group?
Let $A$ be a ring that is finitely generated as an abelian group. Let $I$ be the subgroup of torsion elements. By finite generation, $I$ is finite. On the other hand, $I$ is a two-sided ideal of $A$, and $A/I$ is a ring that is torsion-free as an abelian group. By Borel and Harish-Chandra, $(A/I)^\times$ is finitely-generated. Since $I$ is finite, the map $A^\times \to (A/I)^\times$ is an isogeny (i.e. has finite kernel and cokernel, see Theorem 1.3 in [this paper](https://arxiv.org/abs/1510.02758 "this paper")), and therefore $A^\times$ is finitely generated.


