1
$\begingroup$

I'm just trying to find a reference for the following statement: if $R$ is an $M$-graded integral domain, where $M$ is a monoid, then every unit of $R$ is homogeneous.

This source (in particular, Exercise 1.1) says that the statement is true when $M$ is the group of integers $\mathbb Z$, but I cannot seem to find a source for the more general statement.

Is there a source or simple proof for this proposition?

  • 0
    Notice it's important that the ring be a domain. It's false if there are zero divisors. – 2017-02-25
  • 0
    @MattSamuel yeah, let me edit the question -- I forgot to specify that $R$ is a domain. – 2017-02-25

1 Answers 1

1

This is not true in general. For instance, notice that $R=\mathbb{C}$ can be $\mathbb{Z}/2$-graded by saying that $R_0=\mathbb{R}$ and $R_1=\mathbb{R}i$. But not every unit is homogeneous, since this ring is in fact a field!

More generally, if $K$ is a field and $L$ is an extension of $K$ generated by an element $\alpha$ whose minimal polynomial has the form $x^d-a$, then $L$ has a $\mathbb{Z}/d$-grading whose $n$th graded piece is $K\alpha^n$.