Characterisation of Local Rings
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Theorem
Let $R$ be a ring.
Let $J \lhd R$ be a maximal ideal.
- $(1): \quad$ If the set $R \setminus J$ is precisely the group of units $R^\times$ of $R$, then $\tuple {R, J}$ is a local ring.
The specific interpretation of the notation $\tuple {R, J}$ is unclear. The usual notation for a ring is something like $\struct {R, +, \times}$.
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Proof
$(1): \quad$ Suppose that $J$ is the set of non-units of $R$.
Then by Ideal of Unit is Whole Ring, every ideal not equal to $R$ is contained in $J$.
Therefore $J$ is the unique maximal ideal of $R$, so $\tuple {R, J}$ is local.
$(2): \quad$ Let $x \in R \setminus J$.
Then:
- $J \subsetneq \map I {J \cup \set x} \subseteq R$
where $\map I S$ is the ideal generated by $S$.
Since $J$ is maximal, by definition, $x$ and $J$ generate all of $R$.
Therefore $t x + m = 1$ for some $m \in J$, $t \in R$.
Thus $t x = 1 - m \in 1 + J$ is a unit by hypothesis.
Therefore $x$ is a unit.
Now use part $(1)$.
$\blacksquare$
