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)
so I only need to prove that (A-m)x=A-m
Then I stuck at here .