Editing Jacobson radical (section)

=== Geometric applications ===
{{see also|Nakayama's lemma#Geometric interpretation}}
Although Jacobson originally introduced his radical as a technique for building a theory of radicals for arbitrary rings, one of the motivating reasons for why the Jacobson radical is considered in the commutative case is because of its appearance in [[Nakayama's lemma]]. This lemma is a technical tool for studying [[finitely generated module]]s over commutative rings that has an easy geometric interpretation: If we have a [[vector bundle]] {{nowrap|''E'' → ''X''}} over a [[topological space]] ''X'', and pick a point {{nowrap|''p'' ∈ ''X''}}, then any basis of ''E''|<sub>''p''</sub> can be extended to a basis of sections of {{nowrap|''E''{{!}}<sub>''U''</sub> → ''U''}} for some [[neighborhood (topology)|neighborhood]] {{nowrap|''p'' ∈ ''U'' ⊆ ''X''}}.

Another application is in the case of finitely generated commutative rings of the form <math display="inline">R = k[x_1,\ldots, x_n]\,/\,I </math> for some base ring ''k'' (such as a [[field (mathematics)|field]], or the ring of [[Integer#Algebraic properties|integers]]). In this case the [[Nilradical of a ring|nilradical]] and the Jacobson radical coincide. This means we could interpret the Jacobson radical as a measure for how far the ideal ''I'' defining the ring ''R'' is from defining the ring of functions on an [[algebraic variety]] because of the [[Hilbert Nullstellensatz]] theorem. This is because algebraic varieties cannot have a ring of functions with infinitesimals: this is a structure that is only considered in [[Scheme (mathematics)|scheme theory]].