1
$\begingroup$

Let $F$ be the functor from commutative rings to abelian groups that takes a commutative ring $R$ to its group of units.

I want to show that the functor that takes each group to the group ring ${\bf Z} G$ is left-adjoint to $F$ and am not sure whether this functor has a right adjoint. I think it does not.

  • 0
    What do you know about left adjoint that could help answer the second question? And how are you running into trouble on the first one? You just have to show that group ring maps are the same as homomorphisms into the units, which is pretty straightforward. – 2017-02-05

1 Answers 1

0

Left adjoint functors preserve initial objects. If the functor has a right adjoint, it itself is a left adjoint. But the group of units of the initial object ${\bf Z}$ is $({\pm 1}, \cdot) \cong {\bf Z}/(2)$, a contradiction.