A Note on Jacobson Rings and Polynomial Rings

Miguel Ferrero, Michael M. Parmenter
1989 Proceedings of the American Mathematical Society  
AS is well known, if R is a ring in which every prime ideal is an intersection of primitive ideals, the same is true of R [ X ] . The purpose of this paper is to give a general theorem which shows that the above result remains true when rnany other classes of prime ideals are considered in place of prirnitive ideals. Throughout this paper we assume that R is a ring with identity element and R[X] is the polynomial ring over R in an indeterminate X . A ring R is said to be a Jacobson ring if
more » ... cobson ring if every prime ideal of R is an intersection of primitive (either left or right) ideals. In [7] , Watters proved that if R is a Jacobson ring, the polynomial ring R[X] is also a Jacobson ring. A similar result also holds for Brown-McCoy rings [8], i.e., rings in which every prime ideal is an intersection of maximal ideals. In this note, d will always denote a class of prime rings. We say that an ideal P of R is an M-ideal if R / P E d . When every prime ideal of R is an intersection of d-ideals, the ring R is said to be an M-Jacobson ring. For example, if d is the class of primitive (simple) rings, then an M-Jacobson ring is a Jacobson (Brown-McCoy) ring. The main purpose of this paper is to prove the following Theorem 5. Assume that M is a class of prime rings satisfiing condition (A). If R is an d-Jacobson ring, then so is R [X] . Condition (A) is defined near the beginning of $2. Since primitive (simple) rings satisfy this condition, the above theorem includes as particular cases the results in [7 and 81. However, we show that many other classes of prime rings satisfy condition (A) as well. Some examples include prime Noetherian rings,
doi:10.2307/2046937 fatcat:wzmwfpv6kjaytgb4erpj2n6w3q