TITLE: Consider the ring $R = \mathbb{Q} \times \mathbb{Q}$. Find all the units in $R$.
QUESTION [2 upvotes]: Am I correct in thinking that an element in $R$ takes the form $(a/b, c/d)$, and that each element has a corresponding inverse $(b/a, d/c)$ for all $a, b, c, d \in \mathbb{Z}$ also in $R$? That would mean every element is a unit, and that this is a division ring, right?

REPLY [3 votes]: Because multiplication is applied component-wise, you can simply look at what the units are in $\mathbb{Q}$. An element in $\mathbb{Q}\times\mathbb{Q}$ is a unit, if and only if both its components are units.
And it is $\mathbb{Q}^\times = \mathbb{Q}\setminus\{0\}$ ($\mathbb{Q}^\times$ is the unit group of $\mathbb{Q}$).
So $(\mathbb{Q}\times\mathbb{Q})^\times$ would be $(\mathbb{Q}\setminus\{0\})\times(\mathbb{Q}\setminus\{0\})$.
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