In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of characteristic 0 and let $f(x) \in A[x]$, then:

- $f(x) \in U(A[x])$ $\Leftrightarrow$ $f(0) \in U(A)$ and $f'(x) \in \operatorname{nil}(A[x])$;
- $f(x) \in \operatorname{nil}(A[x])$ $\Rightarrow$ $f'(x) \in \operatorname{nil}(A[x])$;

It is a natural question to ask whether something similar holds if we replace $A$ by a noncommutative unital ring $R$ and consider $R[x]$ ($x$ is assumed to commute with $R$). Of course, this would be much less straightforward because the nilpotent elements of a noncommutative ring in general do not form an ideal. The only result I know of along these lines is the following

**Lemma:** Let $R$ be a (noncommutative) $\mathbb{N}_0$-graded ring and let $r \in R$ be an element of positive degree. Then $1+r$ is a unit iff $r$ is nilpotent.

But I have not been able to trace this result throughout the literature. It is stated in Bass, Connell, and Wright's paper on the Jacobian Problem, but without proof from what I can see, likely because it is not too difficult to prove on one's own. Nevertheless, I would like to have a precise reference for this, because I would like to use it, but would rather avoid including a proof of what appears to be a standard result in noncommutative graded algebra.

Question 1:Do you know of a reference where the above lemma is statedwith a proof?

And more importantly:

Question 2:Are there generalizations of the above lemma, i.e. isanythingknown about units of the form $1 + r_1 + \cdots + r_n$ for $r_1,\dots,r_n \in R$ homogeneous elements of respective degrees $1 \leq d_1 < \dots < d_n$?

I fear this might be too broad of a question or one that is likely impossible to answer in full generality, but I would be happy with some starting places. The motivating example is $U(R[x])$ for $R$ of **characteristic** 0, for which I would like to find out about analogous properties to the aforementioned commutative algebra facts.