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Abstract Algebra

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Moderate

Isomorphism Theorems (Rings): Applying Fourth Isomorphism Theorem

ABSALG-GAJLON

Let $R$ be a ring and $\mathfrak{A}$ an ideal of $R$.

The Fourth Isomorphism Theorem for Rings states that there is a bijection between:

{The ideals $\mathfrak{B}$ of $R$ containing $\mathfrak{A}$} and {The ideals of $R/\mathfrak{A}$} given by $\mathfrak{B}\mapsto \mathfrak{B/A}$

This theorem is also often called the Correspondence Theorem for Rings.

Which of the following is false?

A

Let $\pi: ​R\mapsto S$ be a surjective map of nontrivial commutative rings with identity.
If $\mathfrak{M}$ is a maximal ideal of $S$, then $\pi^{-1}(\mathfrak{M})$ is a maximal ideal of $R$.

B

Let $K$ be a field, and $\pi: K[x]\mapsto K$ the map $\pi(P(x))=P(0)$, for $P\in K[x]$.
Then $xK[x]$ is the only ideal of $K[x]$ containing $\ker(\pi)$.

C

Let $\pi: ​R\mapsto S$ be a surjective map of nontrivial commutative rings with identity.
If $S$ is a field, then $\ker(\pi)$ is a maximal ideal of $R$.

D

The ideal $(5,x)$ is maximal in $\mathbb{Z}[x]$.