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\begin{document}

\title{Math 101: Graduate Linear Algebra \\ Daily Homework \#16}
\date{Assigned Monday, 16 October 2017; due Wednesday, 18 October 2017}

\maketitle

\vspace{1ex}

\begin{problab}{16.1} 
Let $R$ be a commutative ring.  
\begin{enumalph}
\item Let $A,B$ be $R$-algebras (with $1$).  Show that there exists a unique structure of $R$-algebra on the $R$-module $A \otimes_R B$ such that 
\[ (\alpha \otimes \beta)\cdot(\alpha' \otimes \beta') = \alpha\alpha' \otimes \beta\beta' \]
for all $\alpha,\alpha' \in A$ and $\beta,\beta' \in B$.  
\item Let $A$ be an $R$-algebra, let $S$ be a commutative ring, and let $f\colon R \to S$ be a ring homomorphism.  Equip $S$ with the structure of $R$-module via $f$.  (Write out what this means.)  Show accordingly that $A \otimes_R S$ can be given the structure of an $S$-algebra.  
\item Describe the $\C$-algebras $\C \otimes_\R \C$ and $\C \otimes_\C \C$.   
\end{enumalph}
\end{problab}

\end{document}