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===Dual module===
{{see also|Duality (mathematics)#Dual objects}}
The '''[[dual module]]''' of a right ''R''-module ''E'', is defined as {{math|Hom<sub>''R''</sub>(''E'', ''R'')}} with the canonical left ''R''-module structure, and is denoted ''E''<sup>∗</sup>.<ref>{{harvnb|Bourbaki|loc=ch. II §2.3}}</ref> The canonical structure is the [[pointwise]] operations of addition and scalar multiplication. Thus, ''E''<sup>∗</sup> is the set of all ''R''-linear maps {{math|''E'' → ''R''}} (also called ''linear forms''), with operations
<math display="block">(\phi + \psi)(u) = \phi(u) + \psi(u), \quad \phi, \psi \in E^*, u \in E</math>
<math display="block">(r \cdot \phi) (u) = r \cdot \phi(u), \quad \phi \in E^*, u \in E, r \in R,</math>
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