Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand [Wikipedia's](http://en.wikipedia.org/wiki/Jacobson_radical) explanation.



Intuitive Example of a Jacobson Radical
I think my favourite characterization for rings with identity is that y is in the Jacobson radical of R if and only if 1-yx is right invertible for any x in R - so y is sufficiently "zero-like" that moving the unit by its multiples doesn't stop it being invertible.


In fact one can strengthen this to if and only if 1-zyx is actually a unit for any z,x (and deduce from this that the left and right radicals agree).


