Editing Local ring (section)

=== Ring of germs ===

{{main|Germ (mathematics)}}

To motivate the name "local" for these rings, we consider real-valued [[continuous function]]s defined on some [[interval (mathematics)|open interval]] around 0 of the [[real line]]. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an [[equivalence relation]], and the [[equivalence class]]es are what are called the "[[germ (mathematics)|germs]] of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.

To see that this ring of germs is local, we need to characterize its invertible elements. A germ ''f'' is invertible if and only if {{nowrap|''f''(0) ≠ 0}}. The reason: if {{nowrap|''f''(0) ≠ 0}}, then by continuity there is an open interval around 0 where ''f'' is non-zero, and we can form the function {{nowrap|1=''g''(''x'') = 1/''f''(''x'')}} on this interval. The function ''g'' gives rise to a germ, and the product of ''fg'' is equal to 1. (Conversely, if ''f'' is invertible, then there is some ''g'' such that ''f''(0)''g''(0) = 1, hence {{nowrap|''f''(0) ≠ 0}}.)

With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs ''f'' with {{nowrap|1=''f''(0) = 0}}.

Exactly the same arguments work for the ring of germs of continuous real-valued functions on any [[topological space]] at a given point, or the ring of germs of [[differentiable]] functions on any [[differentiable manifold]] at a given point, or the ring of germs of [[rational function]]s on any [[algebraic variety]] at a given point. All these rings are therefore local. These examples help to explain why [[scheme (mathematics)|scheme]]s, the generalizations of varieties, are defined as special [[locally ringed space]]s.