Math 5125 Monday, November 28 November 28, Ungraded Homework Exercise 10.5.9 on page 404 Assume that R is commutative with a 1. (a) Prove that the tensor product of two free R-modules is free. (b) Use (a) to prove that the tensor product of two projective R-modules is projective. (a) First note that R ⊗ R ∼ = R as R-modules. Here is a sketch proof: we have R-maps f : R ⊗ R R → R satisfying f ( r ⊗ s ) = rs , and g : R → R ⊗ R R deﬁned by g ( r ) = r ⊗ 1. Then fg = g f = 1, which shows that f and g are isomorphisms. Using the facts ( A ⊕ B ) ⊗ R C ∼ = ( A ⊗ R C ) ⊕ ( B ⊗ R C ) A ⊗ R ( B ⊕ C ) ∼ = ( A ⊗ R B ) ⊕ ( A ⊗ R C ) , we deduce by induction that R m ⊗ R R n ∼ = R mn . This proves that the tensor product of free modules is free in the ﬁnitely generated case. Actually this argument is still OK if the modules are not ﬁnitely generated. (b) Suppose P , Q are projective R-modules. Then we may write P ⊕ A = E and Q ⊕ B = F for some R-modules
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