A subset T of R is said to be an m-system (generalized multiplicative system) if

(a) P is a prime ideal of R if and only if R - P is an m-system.

(b) Let I be an ideal of R that is disjoint from an m-system T. Show that I is contained in an ideal Q which is maximal respect to the property that Q ∩ T = Ø. Then show that Q is a prime ideal. 

(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. 

(d) Every element c of P(R) is nilpotent.  

Transcribed Image Text:

c,de Tcxde T for some x = R.