# Tagged Questions

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### Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
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### Should $0$ be considered a prime?

Typically, a prime is defined as follows: $p$ is prime iff $(p \mid xy \implies p \mid x$ or $p \mid y)$ and $p$ is not a unit or zero. But for ideals, we say the zero ideal is prime. There is a ...
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### Subring with maximal ideals (prime avoidance). Proof verifying and small question

Let $t∈\Bbb N$ and let $p_1, \dots ,p_t$ be $t$ distinct prime numbers. Show that R = \{α∈\Bbb Q : α = m/n \mbox{ for some } m ∈ \Bbb Z \mbox{ and } n∈\Bbb N \mbox{ such that } n \mbox{ is ...
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### Ideals in Gaussian integers

Let $R:=\mathbb{Z}[i]$. Prove that every nonzero prime ideal $\mathfrak{P}$ of $R$ belongs to one of the following families: 1) $\mathfrak{P}=(1+i)R$ 2) $\mathfrak{P}=(a+bi)R$ where ...
What's the easiest way of showing that; $2\mathbb{Z}\setminus (4n-2)\mathbb{Z}$ is closed under multiplication? (I'm trying to show that $(4n-2)$ is a prime element of $2\mathbb{Z}$ by showing ...