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Commutative Algebra/The lattice of submodules, Noether isomorphism theorems
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Let
m
,
n
∈
N
{\displaystyle m,n\in \mathbb {N} }
. Prove that
(
m
Z
)
/
(
n
Z
)
≅
Z
/
n
gcd
(
m
,
n
)
Z
{\displaystyle (m\mathbb {Z} )/(n\mathbb {Z} )\cong \mathbb {Z} {\big /}{\frac {n}{\gcd(m,n)}}\mathbb {Z} }
.