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Let $I = \langle 1 − ξ \rangle, \xi$ is a primitive $p$-th root of unity, $I$ is a prime ideal of ring of integers $\mathbb Z[\xi]$. Consider the following diagram -

Consider the following diagram -

enter image description here Given, $i$ is the inclusion map, $f$ is a isomorphism and $π$ and $σ$ are the natural quotient maps.

I understand , how both $σ$ and $f ◦ π ◦ i$ are ring homomorphisms taking $1$ to $1$.

But I could not follow they coincide on $\mathbb Z$ and so the diagram is commutative. Can anyone explain or prove in detail pleae?

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A ring morphism $f: \Bbb{Z} \to R$ that preserves the unit is unique:

Indeed, if $n \geq 1$ then $f(n) = f(\sum_{i=1}^n 1) = \sum_{i=1}^n f(1) = \sum_{i=1}^n 1_R = n1_R$ and $f(0) = 0$ and similarly $f(n) = n1_R$ for $n \leq -1$.

We conclude that $f(n) = n 1_R$ for all $n \in \Bbb{Z}$. In your case you have two ring morphisms preserving the unit with domain $\Bbb{Z}$ so they must coincide.

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