# If a finite sum is a unit, then it has a term that is a unit.

Source:

1. Theorem 19.1 (A First Course in Noncommutative Rings by T.Y. Lam)
2. Local Ring on Wikipedia

Theorem 19.1
For any nonzero ring R, the following statements are equivalent:
(1) $$R$$ has a unique maximal left ideal.
(2) $$R$$ has a unique maximal right ideal.
(3) $$R/rad R$$ is a division ring.
(4) $$R$$\ $$U(R)$$ is an ideal of $$R$$.
(5) $$R$$\ $$U(R)$$ is a group under addition.
(6) For any $$n$$, $$a_1+...+a_n\in U(R)$$ implies that some $$a_i\in U(R)$$.
(7) $$a+b\in U(R)$$ implies that $$a\in U(R)$$ or $$b\in U(R)$$.

In the sketch of the proof by Lam, its said that (4)=>(5)=>(6)=>(7) are tautologies.
Let $$R$$ is commutative and $$R$$ is a local ring then its satisfy (1) and (2), how to prove (6)?

• What do you mean by "prove (6)"? The author is not saying that (6) is true in general. They're saying that if one of the 7 statements is true, then so is everything else. Jul 7, 2020 at 13:03
• In particular, a question to ask would be "How to prove (6) $\implies$ (7)?" Jul 7, 2020 at 13:04

Suppose (5) holds and $$\sum a_i\in U(R)$$. Then if all the $$a_i\in R\setminus U(R)$$, it would follow from (5) that $$\sum a_i\notin U(R)$$ since $$\sum a_i\in R\setminus U(R)$$. So apparently one of the $$a_i$$'s has to fall in $$U(R)$$.