I was reading some article on local rings, and it gave the following (equivalent) definitions:
A ring $R$ is $local$ if it satisfies one of the following equivalent properties (so this needs proving)
- $R$ has a unique left maximal ideal.
- $R$ has a unique right maximal ideal.
- $1\neq 0$, and the sum of non-units of $R$ is a non-unit.
- $1\neq 0$, and for all $r\in R$, either $r$ or $1-r$ is a unit.
- If the sum of finite number of elements in $R$ is a unit, then the sum must contain a unit as one of the summands.
Okay, I have no qualm with this definition, except that…
Do we allow the ring to be without an identity?
I mean, what if $1\not\in R$? Can one talk about a local ring that does not have an identity?
Best Answer
Usually, unless otherwise specified, when we say "ring" we mean ring with identity.
In algebraic-geometric or commutative-algebraic contexts, "local ring" usually also means Noetherian.