Is J(R) the Intersection of All Maximal Ideals in R?

[tsang]

Homework Statement

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

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

Homework Equations

I is ideal of ring R
, then I satifies
a+b $\in$ I $\forall$ a,b $\in$ I

ra $\in$ I $\forall$ r $\in$ R

The Attempt at a Solution

I've been trying to use direct definition by having two elements 1-ra, 1-rb $\in$ 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.

 

[Office_Shredder]

1-ra is not in J(R), a is. If a and b are in J(R), 1-ra and 1-rb are units - you need to prove that 1-r(a+b) is a unit as well to show that a+b is contained in J(R)

 

[rachellcb]

Hey I have also been working on this problem- got as far as showing that if u_a and u_b $\in R$ such that u_a(1-ra) =1=(1-ra)u_a
and u_b(1-rb) =1=(1-rb)u then u_bu_a(1-r(a+b))=1.

But (1-r(a+b))u_bu_a=(1-rau_b)_a$\neq?1$

Does anyone have suggestions on how to go from here?

 

[micromass]

It might be easier to show that J(R) is the intersection of all maximal ideals. This is not hard to show.