Editing Standard basis (section)

==Generalizations==
There is a ''standard'' basis also for the ring of [[polynomial]]s in ''n'' indeterminates over a [[field (mathematics)|field]], namely the [[monomial]]s.

All of the preceding are special cases of the [[indexed family]]
<math display="block">{(e_i)}_{i\in I}= ( (\delta_{ij} )_{j \in I} )_{i \in I}</math> 
where <math>I</math> is any set and <math>\delta_{ij}</math> is the [[Kronecker delta]], equal to zero whenever {{nowrap|''i'' ≠ ''j''}} and equal to 1 if {{nowrap|1=''i'' = ''j''}}.
This family is the ''canonical'' basis of the ''R''-module ([[free module]])
<math display="block">R^{(I)}</math>
of all families
<math display="block">f=(f_i)</math> 
from ''I'' into a [[ring (mathematics)|ring]] ''R'', which are [[finite support|zero except for a finite number of indices]], if we interpret 1 as 1<sub>''R''</sub>, the [[Unit (ring theory)|unit]] in ''R''.{{sfn|Roman|2008|p=131|loc=ch. 5}}