a ring modulo its Jacobson radical is semiprimitive

Let $R$ be a ring. Then $J(R/J(R))=(0)$.

Proof:
We will only prove this in the case where $R$ is a unital ring (although it is true without this assumption).

Let $[u]\in J(R/J(R))$. By one of the characterizations of the Jacobson radical, $1-[r][u]$ is left invertible for all $r\in R$, so there exists $v\in R$ such that $[v](1-[r][u])=1$.

Then $v(1-ru)=1-a$ for some $a\in J(R)$. There is a $w\in R$ such that $w(1-a)=1$, and we have $wv(1-ru)=1$.

Since this holds for all $r\in R$, it follows that $u\in J(R)$, and therefore $[u]=0$.

Title	a ring modulo its Jacobson radical is semiprimitive
Canonical name	ARingModuloItsJacobsonRadicalIsSemiprimitive
Date of creation	2013-03-22 12:49:34
Last modified on	2013-03-22 12:49:34
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Theorem
Classification	msc 16N20