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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)

  1. $R$ has a unique left maximal ideal.
  2. $R$ has a unique right maximal ideal.
  3. $1\neq 0$, and the sum of non-units of $R$ is a non-unit.
  4. $1\neq 0$, and for all $r\in R$, either $r$ or $1-r$ is a unit.
  5. 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?

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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.

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  • $\begingroup$ I was actually curious aswell; And I still wonder though: It doesn't follow that it has identity from having a unique maximal ideal (and being commutative) right? i.e. would it still make sense to talk about local (perhaps noncommutative) rings without identity? The definitions seem to vary, including/excluding both identity and commutativity. $\endgroup$ – Christopher.L Sep 18 '18 at 23:00

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