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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#19}
\date{Assigned Monday, 23 October 2017; due Wednesday, 25 October 2017}

\maketitle

\vspace{1ex}

Let $R$ be a commutative ring.

\vspace{2ex}

\begin{problab}{19.1} 
Let $S \subseteq R$ be a multiplicatively closed subset.  Consider the relation $\sim$ on $R \times S$ by $(r,s) \sim (r',s')$ if there exists $x \in S$ such that $x(rs'-r's)=0$.  Show that $\sim$ is transitive.  (What happens if you remove the $x$?)
\end{problab}

\begin{problab}{19.2}
Let $f \in R$ be not nilpotent.  Let $S=\{f^k : k \geq 0\}$.  Consider the ring $R[x]/(fx-1)$, the quotient of the univariate polynomial ring $R[x]$ by the ideal $(fx-1)$.  Show that $R[S^{-1}] \simeq R[x]/(fx-1)$ as rings.  (What happens if $f$ is nilpotent?)
\end{problab}

\end{document}