TITLE: Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain
QUESTION [10 upvotes]: Let $R$ be a commutative domain. 
Prove that the Jacobson radical of $R[X]$, i.e. the intersection of all maximal ideals, is the zero ideal.
Thank you.

REPLY [9 votes]: Exercise 1.4 of Atiyah - Macdonald tells you that in any polynomial ring $R[x]$, the Jacobson radical and nilradical are equal. For your problem let us throw in the additional hypothesis that $R$ is an integral domain. Then the nilradical of $R[x]$ is zero because $R[x]$ is an integral domain and hence the Jacobson radical is zero.