Homework3 - Aaron Goldsmith Homework 3 Section 4.1 Algebra...

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Aaron Goldsmith Homework 3 Section 4.1 Algebra II 4 Let Rx be a unitary cyclic R-module and Define φ : R Rx as r 7→ rx . Then, φ is an R-module homomorphism if R is considered a module over itself: φ ( r + s ) = ( r + s ) x = rx + sx = φ ( r ) + φ ( s ) φ ( rs ) = ( rs ) x = r ( sx ) = ( s ) By a homomorphism theorem, we have Rx = R/ ker φ , and ker φ is a left ideal of R . 9 Let a Ker f Im f with f ( b ) = a . Then, 0 = f ( a ) = f ( f ( b )) = f ( b ) = a and the intersection is trivial. Next, for any a A , we have f ( a - f ( a )) = f ( a ) - f ( f ( a )) = f ( a ) - f ( a ) = 0 and the representation a = ( a - f ( a )) + f ( a ) Ker f + Im f Moreover, this representation is unique for if ( a - f ( a )) + f ( a ) = k + f ( l ) Ker f + Im f then f ( l ) - f ( a ) = a - f ( a ) - k Ker f so f ( l ) - f ( a ) = 0 and f ( l ) = f ( a ). It follows that a - f ( a ) = k . The function ( k,y ) 7→ k + y is then an isomorphism. 11
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This note was uploaded on 08/01/2010 for the course MATH MATH 654 taught by Professor Geller during the Spring '09 term at Texas A&M.

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Homework3 - Aaron Goldsmith Homework 3 Section 4.1 Algebra...

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