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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)?

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    $\begingroup$ 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. $\endgroup$ Jul 7, 2020 at 13:03
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    $\begingroup$ In particular, a question to ask would be "How to prove (6) $\implies$ (7)?" $\endgroup$ Jul 7, 2020 at 13:04

1 Answer 1

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If you mean "prove 6 from 5" then I'll comment on that.

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 $U(R)$.

I guess you did not mean "prove 7 from 6" because 7 is just a special case of 6.

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