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Isomorphism Theorems (Rings): Applying Third Isomorphism Theorem

ABSALG-JLQROT

Let $R$ be a nontrivial ring with identity.

The Third Isomorphism Theorem for Rings states that:

If $\mathfrak{A}$, $\mathfrak{B}$ are ideals of $R$ with $\mathfrak{A}\subseteq \mathfrak{B}$, then $\mathfrak{B}/\mathfrak{A}$ is an ideal of $R/\mathfrak{A}$ and $R/\mathfrak{B} \simeq (R/\mathfrak{A})/(\mathfrak{B}/\mathfrak{A})$.

Let $m$, $n$ be two positive integers such that $n$ divides $m$. Which of the following rings are isomorphic to $\mathbb{Z}/n\mathbb{Z}$, $n\ge 2$.

Select ALL that apply.

A

$(\mathbb{Z}/m\mathbb{Z})/(n\mathbb{Z}/m\mathbb{Z})$

B

$(\mathbb{Z}/n\mathbb{Z})/(m\mathbb{Z}/n\mathbb{Z})$

C

$(\frac1m\mathbb{Z}/\mathbb{Z})/(\frac1n\mathbb{Z}/\mathbb{Z})$

D

$(\frac1n\mathbb{Z}/\mathbb{Z})/(\frac 1m\mathbb{Z}/\mathbb{Z})$

E

$(\frac1m\mathbb{Z}/\mathbb{Z})/(\frac nm\mathbb{Z}/\mathbb{Z})$

F

$(\frac 1n\mathbb{Z}/\mathbb{Z})/(\frac mn\mathbb{Z}/\mathbb{Z})$