Unless otherwise stated a

Let $R$ be a ring and $S$ be a multiplicative subset. Let $M$ be an $R$-module.

(1) Show that the prime ideals in $S^{-1}R$ are in one to one bijection withthe prime ideals in $R$ which do not intersect $S$.

(2) Show that if $R$ is a UFD then $S^{-1}R$ is a UFD if $0 \notin S$.

(3) Let $R$ be a PID and $M$ be a finitely generated torsion free $R$ module. Show that $M$ is a projective $R$-module.

(4) Show that the sequence of $R$-modules $0\to A\to B\to C\to 0$ is exact if after localization at every prime ideal in $R$ it is exact.

(5) Let $R$ be a local ring and $M$ be a finitely generated projective $R$-module then $M$ is a free $R$-module.

(6) Compute $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}$. Is it same as $\mathbb{C}\otimes_{\mathbb{C}}\mathbb{C}$?