Editing Indeterminate (variable) (section)

==As generators==
{{main|Generator (mathematics)}}
Indeterminates are useful in [[abstract algebra]] for generating [[mathematical structure]]s. For example, given a [[field (mathematics)|field]] ''<math>K</math>'', the set of polynomials with coefficients in ''<math>K</math>'' is the [[polynomial ring]] with [[polynomial arithmetic|polynomial addition and multiplication]] as operations. In particular, if two indeterminates ''<math>X</math>'' and ''<math>Y</math>'' are used, then the polynomial ring ''<math>K[X,Y]</math>'' also uses these operations, and convention holds that ''<math>XY=YX</math>''.

Indeterminates may also be used to generate a [[free algebra]] over a [[commutative ring]] ''<math>A</math>''. For instance, with two indeterminates ''<math>X</math>'' and ''<math>Y</math>'', the free algebra ''<math>A\langle X,Y \rangle</math>'' includes sums of strings in ''<math>X</math>'' and ''<math>Y</math>'', with coefficients in ''<math>A</math>'', and with the understanding that ''<math>XY</math>'' and ''<math>YX</math>'' are not necessarily identical (since free algebra is by definition non-commutative).