Given a ring , a left -module , a right -module and a two-sided -module then a map is an -scalar map if
is biadditive, that is and for all and ;
and for all , and .
Such maps can also be called outer linear.
Unlike bilinear maps, scalar maps do not force a commutative multiplication on even when the map is non-degenerate and the modules are faithful. For example, if is an associative ring then the multiplication of , is a -outer linear:
and likewise . Using a non-commutative ring confirms the claim.
It is immediate however that is in fact an -bimodule. This is because:
for all , and . Therefore it is not uncommon to require that indeed all of be an -bimodule.
|Date of creation||2013-03-22 17:24:22|
|Last modified on||2013-03-22 17:24:22|
|Last modified by||Algeboy (12884)|