lecture1

lecture1 - Lecture 1 (Aug 23, 2010) Throughout this lecture...

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Lecture 1 (Aug 23, 2010) Throughout this lecture we assume (unless stated otherwise) that A is a ring (not necessarily comutative) with an identity element 1 0 . Defnition 1. A left A -module over A is a set E with an algebraic structure given as follows: 1. a commutative group law on E (written additively); 2. a scalar multiplication map A × E E, ( a, x ) a · x satisfying the fol- lowing axioms: a ) a · ( x + y ) = a · x + a · y , for all a A, x, y E ; b ) ( a + b ) · x = a · x + b · x , for all a, b A, x E ; c ) a · ( b · x ) = ( ab ) · x , for all a,b A, x E ; d ) 1 · x = x , for all x E . Replacing c) by c ) a · ( b · x ) = ( ba ) · x for all a, b A, x E , we obtain the deFni- tion of a right A -module. Examples: 1. Every commutative group G is a Z -module. 2. If φ : A B is a ring homomorphism, then we obtain an A -module structure on B , with the scalar multiplication given by a · b = φ ( a ) b. In the special case when
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lecture1 - Lecture 1 (Aug 23, 2010) Throughout this lecture...

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