[[Integers]]
# A ring contains $\mathbb{Z}$ or $\mathbb{Z}_{n}$

Let $R$ be a [[ring]]. Then $R$ has a unique [[subring]] isomorphic to [[Integers]] or [[modular arithmetic]] $\mathbb{Z}_{n}$ #m/thm/ring 
given by the image of the unique homomorphism $I : \mathbb{Z} \to R$.

> [!check]- Proof
> Since $I(\mathbb{Z}) \cong \mathbb{Z} / \ker I$ and $\ker I = n \mathbb{Z}$ where $n$ is the [[characteristic]] of $R$.
> <span class="QED"/>

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#state/tidu | #lang/en | #SemBr