# Jacobson ideal

1. Mar 20, 2012

### tsang

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

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)

2. Relevant 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

3. 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 $\in[itex] R, then try to compute x(1-ra), hope can end up format 1-ry, so it can satisfy second condition of being an ideal of ring R, but I still cannot get that format. Unless I haven't use information that 1-ra is unit to help me solve the problem. But not quite sure how to use this bit information. Can anyone please help me with this question? Thanks a lot. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 2. Mar 20, 2012 ### Office_Shredder Staff Emeritus 1-ra is not in J(R), a is. If a and b are in J(R), 1-ra and 1-rb are units - you need to prove that 1-r(a+b) is a unit as well to show that a+b is contained in J(R) 3. Mar 24, 2012 ### rachellcb Hey I have also been working on this problem- got as far as showing that if ua and ub [itex]\in R$ such that ua(1-ra) =1=(1-ra)ua
and ub(1-rb) =1=(1-rb)u then ubua(1-r(a+b))=1.

But (1-r(a+b))ubua=(1-raub)a$\neq?1$

Does anyone have suggestions on how to go from here?

4. Mar 25, 2012

### micromass

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