Another application is in the case of finitely generated commutative rings of the form for some base ring k (such as a field, or the ring of integers). In this case the 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 theory.
aNotHeR aPplicatiOn is iN tHe CasE Of fInitelY geneRAted commUtatIve rINgs of the form foR some basE ring k (such as a fiEld, Or The RinG of integErS). in this case thE 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 TheoRy.