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scalar map (Definition)

Given a ring $ R$, a left $ R$-module $ U$, a right $ R$-module $ V$ and a two-sided $ R$-module $ W$ then a map $ b:U\times V\to W$ is an $ R$-scalar map if

  1. $ b$ is biadditive, that is $ b(u+u',v)=b(u,v)+b(u',v)$ and $ b(u,v+v')=b(u,v)+b(u,v')$ for all $ u,u'\in U$ and $ v,v'\in V$;
  2. $ b(ru,v)=rb(u,v)$ and $ b(u,vr)=b(u,v)r$ for all $ u\in U$, $ v\in V$ and $ r\in R$.

Such maps can also be called outer linear.

Unlike bilinear maps, scalar maps do not force a commutative multiplication on $ R$ even when the map is non-degenerate and the modules are faithful. For example, if $ A$ is an associative ring then the multiplication of $ A$, $ b:A\times A\to A$ is a $ A$-outer linear:

$\displaystyle b(xy,z)=(xy)z=x(yz)=xb(y,z)$
and likewise $ b(x,yz)=b(x,y)z$. Using a non-commutative ring $ A$ confirms the claim.

It is immediate however that $ \langle b(U,V)\rangle$ is in fact an $ R$-bimodule. This is because:

$\displaystyle s(b(u,v)r)=sb(u,vr)=b(su,vr)=sb(u,vr)=(sb(u,v))r$
for all $ u\in U$, $ v\in V$ and $ s,r\in R$. Therefore it is not uncommon to require that indeed all of $ W$ be an $ R$-bimodule.



"scalar map" is owned by Algeboy.
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See Also: bilinear map

Other names:  outer linear
Also defines:  scalar map
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Cross-references: non-commutative, associative, faithful, modules, non-degenerate, even, multiplication, commutative, force, bilinear maps, map, right, ring
There are 3 references to this entry.

This is version 4 of scalar map, born on 2007-07-18, modified 2007-07-18.
Object id is 9778, canonical name is ScalarMap.
Accessed 693 times total.

Classification:
AMS MSC: 13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous)
Pending Errata and Addenda
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