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A local ring is a (say, associative, commutative and unitary) ring $R$ with a unique maximal ideal $\mathfrak{m}$, which in turn determine a uniquely a field $k = R/\mathfrak{m}$. And then my book (and i've seen this in other places as well) says that $(R, \mathfrak{m}, k)$ will denote the local ring.

Silly question, but why do you need to denote a local ring by a triple, isn't all information encompassed in $R$?

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    $\begingroup$ It's useful. That's it. $\endgroup$ – Pedro Tamaroff Jul 31 '17 at 2:28
  • $\begingroup$ ^ that and they're usually setting up notation concisely without having to describe what each of those three symbols stand for. $\endgroup$ – Ravi Jul 31 '17 at 2:30
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If you only write "let $R$ be a local ring" then when you need to talk about the maximal ideal of $R$ you need to also say "let $\mathfrak m$ be the maximal ideal of $R$" because otherwise $\mathfrak m$ isn't defined. Saying "let $(R, \mathfrak m, k)$ be a local ring" lets one introduce three symbols at once and because one often wants to talk about $\mathfrak m$ and $k$ it is useful to have an effective means of introducing them rather than the long winded "let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $k = R/\mathfrak m$".

It's the same reason as one might say at the top of their paper "in this paper $U, V, W$ will denote vector spaces over a fixed field $\mathbf{F}$" rather than having to say "let $V$ be a vector space over a field $\mathbf{F}$" everywhere.

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Another point of view which might be of interest is to look at local rings as a mathematical structure itself.

This notation is then compatible with Bourbaki's notation (Elements of Mathematics vol. 1) and further emphasises the importance of a that a morphism between local rings should preserve their maximal ideals (local homomorphism of rings) and by simple ring theory pass to the quotient to a morphism on their residue field.

In other words a morphism between $(A, \mathfrak m, k)$ and $(B, \mathfrak n, l)$ is a triple of morphisms (one of non unital rings and two unitary) $f,\phi, \overline\phi$ such that the natural diagram

commutative diagram

commutes.

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