Module Fundamentals

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Chapter 4Module Modules and Definitions and CommentsA vector spaceMover a fieldRis a set of objects called vectors, which can be added,subtracted and multiplied by scalars (members of the underlying field). ThusMis anabelian group under addition, and for eachr Randx Mwe have an elementrx multiplication is distributive and associative, and the multiplicative identity of thefield acts as an identity on vectors. Formally,r(x+y)=rx+ry;(r+s)x=rx+sx;r(sx) =(rs)x;1x=xfor allx,y Mandr,s R. A Module is just a vector space over a ring. The formaldefinition is exactly as above, but we relax the requirement thatRbe a field, and insteadallow an arbitrary ring. We have written the productrxwith the scalarron the left, andtechnically we get aleftR-moduleover the ringR. The axioms of arightR-moduleare(x+y)r=xr+yr;x(r+s)=xr+ xs;(xs)r=x(sr),x1=x.

4.1. MODULES AND ALGEBRAS 3 3. IfRisacommutativering,thenM n(R),thesetofalln× nmatriceswithentries inR,isanR-algebra(seeExample4of(4.1.3)). 4. IfRisacommutativering ...

  Algebra