cmi-entrance 2015 Q6

cmi-entrance · India · pgmath 4 marks Not Maths
Let $R$ be an integral domain such that every non-zero prime ideal of $R[X]$ (where $X$ is an indeterminate) is maximal. Choose the correct statement(s):
(A) $R$ is a field;
(B) $R$ contains $\mathbb{Z}$ as a subring;
(C) Every ideal in $R[X]$ is principal;
(D) $R$ contains $\mathbb{F}_{p}$ as a subring for some prime number $p$.
Let $R$ be an integral domain such that every non-zero prime ideal of $R[X]$ (where $X$ is an indeterminate) is maximal. Choose the correct statement(s):\\
(A) $R$ is a field;\\
(B) $R$ contains $\mathbb{Z}$ as a subring;\\
(C) Every ideal in $R[X]$ is principal;\\
(D) $R$ contains $\mathbb{F}_{p}$ as a subring for some prime number $p$.