Abstract Algebra

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Isomorphism Theorems (Rings): Second Isomorphism Theorem, Integer


Let $R$ be a nontrivial ring with identity, and let $\mathfrak{A}$, $\mathfrak{B}$ be ideals in $R$.

The Second Isomorphism Theorem for Rings states that $(\mathfrak{A}+\mathfrak{B})/\mathfrak{B}\simeq\mathfrak{A}/(\mathfrak{A}\cap\mathfrak{B})$

For $m, n\ge1$, let $R=\mathbb{Z}$, the ring of integers, and let $\mathfrak{A}=m\mathbb{Z}$, $\mathfrak{B}=n\mathbb{Z}$.

(1) $\mathfrak{A}+\mathfrak{B}$ equals


(2) $\mathfrak{A}\cap\mathfrak{B}$ equals to


(3) $(\mathfrak{A}+\mathfrak{B})/\mathfrak{B}$ equals


(4) $\mathfrak{A}/(\mathfrak{A}\cap\mathfrak{B})$ equals