Editing Associative algebra (section)

== Separable algebra ==
{{main|Separable algebra}}

Let ''A'' be an algebra over a commutative ring ''R''. Then the algebra ''A'' is a right{{efn|Editorial note: as it turns out, ''A''<sup>e</sup> is a full matrix ring in interesting cases and it is more conventional to let matrices act from the right.}} module over {{nowrap|1=''A''<sup>e</sup> := ''A''<sup>op</sup> ⊗<sub>''R''</sub> ''A''}} with the action {{nowrap|1=''x'' ⋅ (''a'' ⊗ ''b'') = ''axb''}}. Then, by definition, ''A'' is said to [[separable algebra|separable]] if the multiplication map {{nowrap|''A'' ⊗<sub>''R''</sub> ''A'' → ''A'' : ''x'' ⊗ ''y'' ↦ ''xy''}} splits as an ''A''<sup>e</sup>-linear map,{{sfn|Cohn|2003|loc=§ 4.7|ps=none}} where {{nowrap|''A'' ⊗ ''A''}} is an ''A''<sup>e</sup>-module by {{nowrap|1=(''x'' ⊗ ''y'') ⋅ (''a'' ⊗ ''b'') = ''ax'' ⊗ ''yb''}}. Equivalently,{{efn|To see the equivalence, note a section of {{nowrap|''A'' ⊗<sub>''R''</sub> ''A'' → ''A''}} can be used to construct a section of a surjection.}}
''A'' is separable if it is a [[projective module]] over {{nowrap|''A''<sup>e</sup>}}; thus, the {{nowrap|''A''<sup>e</sup>}}-projective dimension of ''A'', sometimes called the '''bidimension''' of ''A'', measures the failure of separability.