Can anyone explain what a Jacobson radical is using an intuitive example? I can't quite understand Wikipedia's explanation.
I think my favourite characterization for rings with identity is that y is in the Jacobson radical of R if and only if 1yx is right invertible for any x in R  so y is sufficiently "zerolike" that moving the unit by its multiples doesn't stop it being invertible. In fact one can strengthen this to if and only if 1zyx is actually a unit for any z,x (and deduce from this that the left and right radicals agree). 


Lucky timing, I just worked out a couple of examples for the class I'm teaching this semester. Let A be a commutative local ring, m the maximal ideal, and k=A/m the residue field. Let's consider two rings: M = M_{2}(A) (2by2 matrices with entries in A), and I = the Iwahori order consisting of 2by2 matrices with entries in A whose lowerleft entry is in m, i.e. matrices of the form
We'll compute rad(M) and rad(I) using the fact that rad = intersection of annihilators of simple left modules = { x : 1  xy is a unit for all y }. First let's compute rad(M). M acts naturally on k^2 by left multiplication (via M_{2}(A) >> M_{2}(k)), and this is a simple Mmodule. The annihilator of k^2 is M_{2}(m) = 2by2 matrices with entries in m, so rad(M) is contained in M_{2}(m). On the other hand every element x in M_{2}(m) has the property that 1xy is a unit in M_{2}(A) for all y in M_{2}(A) (since xy is in M_{2}(m), the determinant of 1xy is a unit); therefore rad(M) contains M_{2}(m), and we've shown rad(M) = M_{2}(m). Next let's compute rad(I). Now k is a simple left Imodule in two ways: first by multiplication by the upperleft entry (mod m), and second by multiplication by the lowerright entry (mod m). (Check that if x,y are in I then the upperleft entry of xy is congruent mod m to the product of the upperleft entries of x and y, and similar for the lower right.) The annihilator of the first Imodule is therefore matrices of the form



Well, if R is a finitely generated commutative ring then J(R) is just the nilradical, so for example Z[x]/(x^2) has Jacobson radical (x). The intuition I have about the nilradical (and by extension, the Jacobson radical) is that it measures how far R is from behaving like the ring of functions on a space. More precisely, if R = C[x_{1}, .. x_{n}]/I is a finitelygenerated Calgebra with corresponding variety V, then the ring of functions on V is C[x_{1}, ... x_{n}]/rad(I) = R/Nil(R) by the Nullstellensatz. This is because elements of the nilradical of R, which is rad(I), are nilpotent on V, and an honesttogoodness function on a space can't be nilpotent unless it's identically zero. 

