# A basic question about rings

Perhaps this is a trivial question, but I have no idea how to justify it.

Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and multiplication $(+, \ast)$ say satisfying the condition that $(R, +)$ is isomorphic to $G_1$ and the group of units of $R$, say $U(R)$, satisfy $(U(R), \ast) \cong G_2$.

For instance, the existence of finite fields of order $p^n$ for a prime $p$ implies that the pair $(G_1, G_2)$ with $G_1$ being the group $(C_p)^n = C_p \times \cdots \times C_p$ and $G_2$ being the cyclic group of order $p^n - 1$ is a ring-compatible pair.

Are all pairs of groups $(G_1, G_2)$ with $G_1$ abelian and if $G_1, G_2$ are finite groups, $|G_1| > |G_2$, ring compatible?

I apologize in advance if this question is trivial.

• Yes that is a useful remark; thank you! – Stanley Yao Xiao Jan 17 '14 at 16:59
• But not even every cyclic group is the unit group of some ring. – Nathan Jan 17 '14 at 17:01
• Also, if $G_1$ is or prime order, then there is only one choice for $R$, and thus for $G_2$. – Pierre-Guy Plamondon Jan 17 '14 at 17:03
• Let's extend this question to infinite group: for instance Is the pair $(\mathbb{C}, \mathbb{R})$ ring compatible? – Ali Taghavi Jan 17 '14 at 23:08
• According to Kadison conjecture a related question can be as follows: Let G be a torsion free group, Is $\mathbb{C}G, G$ ring compatible? – Ali Taghavi Jan 17 '14 at 23:11

The answer for your question is NO, not every pair of groups is ring compatible, even if the first group is abelian. There are abelian groups $G_1$ (called Nil groups), in which only zero multiplication turns $G_1$ into a ring. A complete discussion of your question can be find in "L. Fuchs, Infinite abelian groups" (Despite the name of book, you can find facts about additive group of finite rings)