(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let R be a unital ring. Define J(R)={a [itex]\in[/itex] R| 1-ra is a unit for any r [itex]\in[/itex] R}

Show that J(R) is an ideal in R. (It is called the Jacobson ideal of R)

2. Relevant equations

I is ideal of ring R

, then I satifies

a+b [itex]\in[/itex] I [itex]\forall[/itex] a,b [itex]\in[/itex] I

ra [itex]\in[/itex] I [itex]\forall[/itex] r [itex]\in[/itex] R

3. The attempt at a solution

I've been trying to use direct definition by having two elements 1-ra, 1-rb [itex]\in[/itex] J(R), then I tried to do (1-ra)+(1-rb) and hope to end up another element which has format 1-rc, but I couldn't get it.

Similarly, I let some x [itex]\in[itex] R, then try to compute x(1-ra), hope can end up format 1-ry, so it can satisfy second condition of being an ideal of ring R, but I still cannot get that format.

Unless I haven't use information that 1-ra is unit to help me solve the problem. But not quite sure how to use this bit information.

Can anyone please help me with this question? Thanks a lot.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Jacobson ideal

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