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proposition: x∈The Jacobson radical <=> 1-xy is a unit in commutative ring A for all y∈A
I have proved (=>)
I don't figure out a detail of the proof of (<=).
Here is the proof on book:
suppose x∉m for some maximal ideal m.
then m and x generate the unit ideal (1). (*Why?*It's the only thing I confused about in this proof)
.......
Here is what I thought:
(m) is still (m)
(x)=Ax=mx+(A-m)x=m+(A-m)x
so I only need to prove that (A-m)x=A-m
Then I stuck at here .
By definition $\mathfrak{m} \subset A$ is maximal if and only if for all other ideals $I$ for which $ \mathfrak{m} \subseteq I \subseteq A$ holds, either $I = \mathfrak{m}$ or $I = A$.
The ideal $(\mathfrak{m}, x)$, generated by $\mathfrak{m}$ and $x$ indeed sits in between $\mathfrak{m}$ and $A$ as above, i.e. $$ \mathfrak{m} \subseteq (\mathfrak{m}, x) \subseteq A. $$ However, it cannot be equal to $\mathfrak{m}$, as $x \not \in \mathfrak{m}$. Therefore $(\mathfrak{m},x) = A$.
