Editing Linear form (section)

== Over a ring ==
[[Module (mathematics)|Modules]] over a [[Ring (mathematics)|ring]] are generalizations of vector spaces, which removes the restriction that coefficients belong to a [[Field (mathematics)|field]].  Given a module {{mvar|M}} over a ring {{mvar|R}}, a linear form on {{mvar|M}} is a linear map from {{mvar|M}} to {{mvar|R}}, where the latter is considered as a module over itself. The space of linear forms is always denoted {{math|Hom<sub>''k''</sub>(''V'', ''k'')}}, whether {{mvar|k}} is a field or not. It is a [[right module]] if {{mvar|V}} is a left module.

The existence of "enough" linear forms on a module is equivalent to [[Projective module|projectivity]].<ref>{{Cite book|last=Clark|first=Pete L.|url=http://alpha.math.uga.edu/~pete/integral2015.pdf|title=Commutative Algebra|publisher=Unpublished|at=Lemma 3.12}}</ref>  
{{math theorem|math_statement=An {{mvar|R}}-[[Module (mathematics)|module]] {{mvar|M}} is [[projective module|projective]] if and only if there exists a subset <math>A\subset M</math> and linear forms <math>\{f_a\mid a \in A\}</math> such that, for every <math>x\in M,</math> only finitely many <math>f_a(x)</math> are nonzero, and
<math display="block">x=\sum_{a\in A}{f_a(x)a}</math>
|name=Dual Basis Lemma
}}