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