Editing Commutative ring (section)

=== Finite generation ===
An ''R''-algebra ''S'' is called [[finitely generated algebra|finitely generated (as an algebra)]] if there are finitely many elements ''s''<sub>1</sub>, ..., ''s''<sub>''n''</sub> such that any element of ''s'' is expressible as a polynomial in the ''s''<sub>''i''</sub>. Equivalently, ''S'' is isomorphic to
{{block indent|1= ''R''[''T''<sub>1</sub>, ..., ''T''<sub>''n''</sub>] / ''I''. }}

A much stronger condition is that ''S'' is [[finitely generated module|finitely generated as an ''R''-module]], which means that any ''s'' can be expressed as a ''R''-linear combination of some finite set ''s''<sub>1</sub>, ..., ''s''<sub>''n''</sub>.