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Find all the ring homomorphic images of the integers


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.3

Find all homomorphic images of $\mathbb{Z}$.


Solution: Recall that every additive subgroup of $\mathbb{Z}$ is cyclic, and moreover the additive subgroups of $\mathbb{Z}$ are precisely $\langle k \rangle$ for each $k \in \mathbb{Z}$.

Note that for each $ka \in \langle k \rangle$ and $m \in \mathbb{Z}$, $mka = kma \in \langle k \rangle$ and $kam \in \langle k \rangle$, so that $\langle k \rangle$ is a two sided ideal of $\mathbb{Z}$.

Thus the ideals of $\mathbb{Z}$ are precisely of the form $\langle k \rangle$ for each $k \in \mathbb{Z}$. By the First Isomorphism Theorem for rings, the homomorphic images of $\mathbb{Z}$ are precisely $\mathbb{Z}/\langle k \rangle$ for each $k \in \mathbb{Z}$.


Linearity

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