scalar map

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×VW is an R-scalar map if

  1. 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,uU and v,vV;

  2. 2.

    b(ru,v)=rb(u,v) and b(u,vr)=b(u,v)r for all uU, vV and rR.

Such maps can also be called outer linear.

Unlike bilinear maps, scalar maps do not force a commutativePlanetmathPlanetmathPlanetmath multiplication on R even when the map is non-degenerate and the modules are faithfulPlanetmathPlanetmath. For example, if A is an associative ring then the multiplication of A, b:A×AA is a A-outer linear:


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

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


for all uU, vV and s,rR. Therefore it is not uncommon to require that indeed all of W be an R-bimodule.

Title scalar map
Canonical name ScalarMap
Date of creation 2013-03-22 17:24:22
Last modified on 2013-03-22 17:24:22
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 7
Author Algeboy (12884)
Entry type Definition
Classification msc 13C99
Synonym outer linear
Related topic BilinearMap
Defines scalar map