Editing Glossary of ring theory (section)

== F ==
{{glossary}}

{{term|1=field}}
{{defn|1=A [[Field (mathematics)|field]] is a commutative division ring; i.e., a nonzero ring in which each nonzero element is invertible.}}

{{term|1=filtered ring}}
{{defn|1=A [[filtered ring]] is a ring with a filtration.}}

{{term|1=finitely generated}}
{{defn|no=1|1=A left ideal ''I'' is ''[[finitely generated ideal|finitely generated]]'' if there exist finitely many elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} such that {{nowrap|1=''I'' = ''Ra''<sub>1</sub> + ... + ''Ra''<sub>''n''</sub>}}. A right ideal ''I'' is ''finitely generated'' if there exist finitely many elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} such that {{nowrap|1=''I'' = ''a''<sub>1</sub>''R'' + ... + ''a''<sub>''n''</sub>''R''}}. A two-sided ideal ''I'' is ''finitely generated'' if there exist finitely many elements {{nowrap|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} such that {{nowrap|1=''I'' = ''Ra''<sub>1</sub>''R'' + ... + ''Ra''<sub>''n''</sub>''R''}}.}}
{{defn|no=2|A '''[[finitely generated ring]]''' is a ring that is finitely generated as '''Z'''-algebra.}}

{{term|1=finitely presented}}
{{defn|1=A finitely presented algebra over a commutative ring ''R'' is a (commutative) [[associative algebra]] that is a [[quotient ring|quotient]] of a [[polynomial ring]] over ''R'' in finitely many variables by a [[finitely generated ideal]].<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.4.1}}</ref><!-- the non-commutative case should be discussed. -->}}

{{term|1=free}}
{{defn|no=1|1=A [[free ideal ring]] or a fir is a ring in which every right ideal is a free module of fixed rank.}}
{{defn|no=2|1=A semifir is a ring in which every finitely generated right ideal is a free module of fixed rank.}}
{{defn|no=3|1=The [[free product of associative algebras|free product]] of a family of associative is an associative algebra obtained, roughly, by the generators and the relations of the algebras in the family. The notion depends on which category of associative algebra is considered; for example, in the category of commutative rings, a free product is a tensor product.}}
{{defn|no=4|1=A [[free ring]] is a ring that is a [[free algebra]] over the integers.}}

{{glossary end}}