For a ring $A$ with $A^*\cong \Bbb{Z}$ show that $1+1=0$ Let $A$ be a ring and denote $A^*:=\{x\in A|\exists y:xy=1\}$. Suppose $A^*\cong \Bbb{Z}$, show that $1+1=0$.
I will assume I am to show that $1_{A}+1_A=0$ but I am not quite sure what assumption I may make based on the isomorphism (A groups one I presume).
$A^*$ is a group, there is not doubt about it. $\Bbb{Z}$ is an additive group. I guess that by the definition of isomorphism, there exists $\phi:\Bbb{Z}\to A^*$ such that $\phi(0)=1_A$ (Is it true? How can I formally justify that?). If it is true, then $\phi(0+0)=\phi(0)=1_A+1_A=1_A$ meaning $1_A=0$ (I suppose). I am quite new to Ring Theory. Is there a point in my approach? How should I approach it?
 A: If $A$ is any (unital) ring, there is a unique ring homomorphism $\mathbb{Z} \to A$ sending $1_{\mathbb Z} \mapsto 1_A$. If the kernel of this map is $n \mathbb{Z}$ then the universal property of quotients gives an embedding $\mathbb Z/n\mathbb Z \hookrightarrow A$. It follows that there is a group embedding $(\mathbb Z/n \mathbb Z)^* \hookrightarrow A^*$. First of all, if $n = 1$ then $1_A = 0_A$ so we can rule out that case. Otherwise, if $n \ne 2$ you can show that it is impossible to embed $(\mathbb Z/n\mathbb Z)^*$ into $\mathbb Z$ since the former has torsion and the latter is torsion-free.
A: An even simpler argument, without the need of the homomorphism $\Bbb Z \rightarrow A$:
We have $(-1)_A^2 = 1_A$, which means that $(-1)_A$ is a torsion element in $A^*$. But by assumption, $A^* \simeq \Bbb Z$ is torsion-free, thus the only torsion element in $A^*$ is $1_A$.
This means $(-1)_A = 1_A$ and hence $1_A + 1_A = 0_A$.

On the other hand, it would be better to show that there exists a ring $A$ with $A^* \simeq \Bbb Z$: otherwise it would be equally true to say that $A^* \simeq \Bbb Z \implies 3_A = 0$.
The ring $A = \Bbb F_2[X, Y]/(XY - 1)$ is such an example. It is sometimes more convenient to denote this ring by $\Bbb F_2[X, X^{-1}]$. Under this notation, it is easy to see that $A^*$ is infinite cyclic with generator $X$.
