(a) If I is an ideal of R, then P(I) = I ∩ P(R). In particular, P(P(R)) = P(R). 

(b) P(R) is the smallest ideal K of R such that P(R/K) = 0. In particular, P(R/P(R)) = 0, whence R/P(R) is semi prime. 

(c) An ideal I is said to have the zero property if every element of I has the zero property (Exercise 1 (c)). Show that the zero property is a radical property (as defined in the introduction to Section 2), whose radical is precisely P(R).

Data from exercise 1(c)

An element r of R is said to have the zero property if every m-system that contains r also contains 0. Show that the prime radical P(R) is the set M of all elements of R that have the zero property.