f09_mthsc852_hw18

f09_mthsc852_hw18 - (4) If R is a commutative ring with 1...

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MTHSC 851/852 (Abstract Algebra) Dr. Matthew Macauley HW 18 Due Monday, December 7th, 2009 (1) (a) Suppose M is an R -module, that x,y Tor( M ) with | x | = r , | y | = s , and that r and s are relatively prime in R . Show that | x + y | = rs . (b) Give an example of an R -module M over a commutative ring R where Tor( M ) is not a submodule. (2) Suppose R is a commutative ring and M is an R -module. A submodule N is called pure if rN = rM N for all r R . (a) Show that any direct summand of M is pure. (b) If M is torsion free and N is a pure submodule show that M/N is torsion free. (c) If M/N is torsion free show that N is pure. (3) If R is a commutative ring with 1 and M is an R -module define a function φ : x ˆ x from M to its double dual M ** = ( M * ) * by setting ˆ x ( f ) = f ( x ), all f M * . Show that φ is an R -homomorphism. Under what circumstances is φ a monomorphism?
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Unformatted text preview: (4) If R is a commutative ring with 1 and x 1 , x 2 are distinct indeterminantes show that R [ x 1 ,x 2 ] and R [ x 1 ] R R [ x 2 ] are isomorphic as R-algebras. (5) Suppose A is a nitely generated abelian group. (a) Compute A Z Q . (b) Dene f : A A Z Q by setting f ( a ) = a 1 for all a A . Show that f is a homomorphism. Under what circumstances if f a monomorphism? (6) If A is an abelian group show that Z n Z A = A/nA . (7) If K M N 0 is an exact sequence of left R-modules and L is a right R-module show that L R K L R M L R N 0 is an exact sequence of abelian groups....
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