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 ﬁnitely generated abelian group. (a) Compute A ⊗ Z Q . (b) Deﬁne 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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This note was uploaded on 03/11/2012 for the course MTHSC 851 taught by Professor Staff during the Fall '08 term at Clemson.
- Fall '08