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\begin{document}

\title[101 Homework \#20]{Math 101: Algebra I \\ Homework, Day \#20}
\date{Assigned Friday, 14 October 2016; due Monday, 17 October 2016}

\maketitle

\vspace{1ex}

\begin{problab}{JV20.A} 
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 $A \otimes_R B$ with the property 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 and let $f:R \to S$ be a ring homomorphism.  Give $S$ the structure of $R$-module via $f$.  Show 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}