The nil radical N(R) of R is the ideal generated by the set of all nil ideals of R.

(a) N(R) is a nil ideal.

(b) N(N(R)) = N(R).

(c) N(R/ N(R)) = 0.

(d) P(R) ⊂ N(R) ⊂ J(R).

(e) If R is left Artinian, P(R) = N(R) = J(R).

(f) If R is commutative P(R) = N(R).