What is $\Bbb R^{\times}$? [unit group, ring to "times" power] I'm doing some sheets for my Abstract Algebra class and I can't seem to remember the group defined as $\mathbb{R}^{\times}$. It's obviously some variation of $\mathbb{R}$ but I'm away from college on reading week so can't ask my tutor. If someone could clear up the confusion I'd be grateful.
This is the usual notation for the unit group of the ring $R$, that is, the group of invertible elements using ring multiplication as the group operation. Note that this is usually not the same thing as $R \setminus \{0\}$, because most elements aren't invertible (if it is the same, your ring is a field, by definition). While you can talk about $R \setminus \{0\}$ under multiplication, it's usually just a monoid (or even a semigroup, if you permit rings without unity).
The notation is often used on the form $\Bbb R^*$ i.e. with a star and it means $\Bbb R\setminus \{0\}$ and we have $(\Bbb R^*,\times)$ is a multiplicative  group.