cmi-entrance 2025 Q9

cmi-entrance · India · pgmath 4 marks Groups Ring and Field Structure
Pick the correct statement(s) from below.
(A) There exists a maximal ideal $M$ of $\mathbb { Z } [ x ]$ such that $M \cap \mathbb { Z } = ( 0 )$.
(B) If $M$ is a maximal ideal of $\mathbb { Z } [ x ]$, then $\mathbb { Z } [ x ] / M$ is finite.
(C) If $I$ is an ideal of $\mathbb { Z } [ x ]$ such that $\mathbb { Z } [ x ] / I$ is finite, then $I$ is maximal.
(D) The ideal $\left( 7 , x ^ { 2 } - 14 x - 2 \right)$ in $\mathbb { Z } [ x ]$ is maximal.
Pick the correct statement(s) from below.\\
(A) There exists a maximal ideal $M$ of $\mathbb { Z } [ x ]$ such that $M \cap \mathbb { Z } = ( 0 )$.\\
(B) If $M$ is a maximal ideal of $\mathbb { Z } [ x ]$, then $\mathbb { Z } [ x ] / M$ is finite.\\
(C) If $I$ is an ideal of $\mathbb { Z } [ x ]$ such that $\mathbb { Z } [ x ] / I$ is finite, then $I$ is maximal.\\
(D) The ideal $\left( 7 , x ^ { 2 } - 14 x - 2 \right)$ in $\mathbb { Z } [ x ]$ is maximal.