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Problem 1 (DF, problem 6, p. 375)

Let $R$ be an integral domain with field of fractions $Q$. Prove that $(Q/R)\otimes_{R} (Q/R)=0.$ In particular $\Q/\Z\otimes\Q/\Z=0$.

Problem 2 (DF, problem 5, p. 375)

Let $G$ be a finite abelian group and let $p^{k}$ be the highest power of a prime $p$ dividing $\vert G\vert$. Prove that $Z/p^{k}\Z\otimes G$ is the Sylow $p$-subgroup of $G$.


Prove that, if $M$ is a module over an integral domain $R$, and $Q$ is the quotient field of $R$, then the rank of $M$ is the dimension of the $Q$-vector space $Q\otimes M$.