# Relation(s) between units and nilpotent elements in graded noncommutative rings

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:

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

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 stated with a proof?

And more importantly:

Question 2: Are there generalizations of the above lemma, i.e. is anything known 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.

• Your "standard facts " are not correct. In $\mathbb{F}_p[x]$, $1+x^p$ satisfies the hypotheses of 1. but is not invertible, and $g(x)=x^p$ is not nilpotent though $g'(x)=0$.
– abx
Commented Dec 16, 2022 at 7:34
• However, you can easily fix (1) either by replacing $f'$ by $f-f(0)$, or by making it a one-sided implication "$\Rightarrow$" like in (2). Commented Dec 16, 2022 at 9:23
• @abx: My apologies, I implicitly meant characteristic 0. I will fix that immediately.
– M.G.
Commented Dec 16, 2022 at 12:48
• In your lemma you don't need $r$ to have positive degree. Rather, you need the grade-$0$ component of $r$ to be zero. For question 2, you might look at some of the work Agata Smoktunowicz did on graded nil rings. Really weird things can happen. Commented Dec 16, 2022 at 16:08
• @PaceNielsen: Thanks for the reply! I basically copied the lemma from the BCW's paper. If I understand you correctly, you are saying that the non-trivial direction of the equivalence in the lemma also holds for non-homogeneous elements $r$ as long as all the components are of positive degree?
– M.G.
Commented Dec 16, 2022 at 16:14

The lemma for homogeneous $$r$$ is easy, so that there is no needs for special reference.
Proof: Let $$(1+r)(1+y_1+\dots +y_n)=1$$ in an associative graded ring, where $$\deg r = d>0$$, $$\deg y_i=i$$, $$y_i$$ are homogeneous. Then $$y_i+ry_{i-d}=0$$ for all $$i$$ (we assume here $$y_0 =1$$ and $$y_t=0$$ for $$t$$ not in $$[0,n]$$), so, the only nonzero $$y_i$$ with $$i>0$$ are $$y_d = -r$$, $$y_{2d } =r^2$$, etc. Moreover, for the last nonzero $$y_{md} = \pm r^d$$ we get $$0 = r y_{md} = \pm r^{m+1}$$.