I'm learning about localizations, and I came across this statement: $$ \left(\mathbb{Z}/(4)\right)_{(2)} = \mathbb{Z}/(4) $$ Now, this statement makes sense because inverting all of the odd elements have no effect since the images of odd elements in $\mathbb{Z}/(4)$ are all invertible. But I'm struggling to write this down formally, what exactly can I write down to convince myself of this fact beyond a reasonable doubt?
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The complement of $(2)$ is the set of all units of $\mathbb{Z}/(4)$.
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So, if $I$ is an ideal of $R$ and $P$ is a prime ideal of $R$, then when we write $(R/I)_{P}$ do we actually mean the homomorphic image of $P$ under the quotient map? – klein4 Mar 01 '21 at 16:41
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Probably, since $P$ doesn't live in $(R/I) $. – Nektarios Orfanoudakis Mar 01 '21 at 16:43