- I know, that the multiplicative group of $\mathbb{R}$ is create on the set $\mathbb{R}\setminus \{0\}$. But how we can multiply real numbers on the $0$ after this?
This point was unswered, I think. But for now, I cannot understand, why we can say that $\mathbb{R}$ -- field. I think that field -- is a $\mathbb{R}\setminus\{0\}$. Becouse in $\mathbb{R}$ exist non-unit element.
- Is the equation $ax = b$ unsolvable if $\forall x\in\mathbb{R}$ and $a,b = 0$ or this equation has infinite number of roots (the all real numbers)? I think the first (becouse $0$ doesn't unit element in multiplication group), but not sure.
Hint: For each real number $a\ne 0$, $a0 = 0 = 0a$, since $a0 = a(a+(-a)) = a^2 - a^2=0$. Similar for $0a$. So $0$ is absorbing. It holds for every ring.