Let $R$ be a ring or more genrally a $R$-module $M$. My question is what is the intuition behind or more precisely when following argumentation is applicable:
Let $\mathcal{P}$ be a ring theoretical property then
$R$ has $\mathcal{P}$ if and only if ALL localisations $R_m$ with respect to the maximal ideals $m$ of $R$ have property $\mathcal{P}$
What is the deeper nature of the properties $\mathcal{P}$ which are compatible with the argumentation technique described above? Taking into account the theory of schemes one can simply say that this concerns "local" properties. Is there any intuition to recognize when such property has local nature?
Here, the intuition is indicated by terminology. Let us call "local property" a property that is true if and only it is true for all localizations at maximal ideals. Then, a local property is a property that, translated (see below) in terms of algebraic varieties, is true for an algebraic variety if and only if it is true at each point of the variety.
For example, being a regular ring is the algebraic translation of being a regular variety. By definition, a variety is regular if it does not have any singular point. There are various definitions of a singular point, which are all equivalent with the fact that the local ring at this point is not a regular local ring.
This translation from geometry to algebra, and conversely from algebra to geometry started with Hilbert's Nullstellensatz that establishes a correspondence between affine subvarieties and prime ideals in a polynomial ring, but its richness has been completely been understood only with Grothendieck's scheme theory.
In short, an affine algebraic set is defined as the common zeros of an ideal $I$, to which is associated the ring $R =K[x_1, \ldots, x_n]/I$. This ring is the ring of polynomial functions on the algebraic set. The local ring of $R$ at a maximal ideal $m$ is the ring of polynomial functions on the variety that are defined in a neighbourhood of the point associated to $m$. The idea of Grothendieck was to push to its limit this translation to algebra, and to reverse it by defining an algebraic variety (more precisely an ''affine scheme'') by a ring together with its Zariski topology, and the ringed space of local rings. In this context, the points are not only the maximal ideals but all prime ideals (corresponding to sub varieties).
As this theory applies to any commutative ring, and not only on those that are associated to algebraic varieties, one needs often to assume further properties for proving that a property is a local property (for example that the ring is Noetherian, or is an algebra that is finitely generated over a field).