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\title[101 Homework \#15]{Math 101: Algebra I \\ Homework, Day \#15}
\date{Assigned Wednesday, 5 October 2016; due Friday, 7 October 2016}

\maketitle

\begin{problab}{JV15.A}
Let $R$ be a commutative ring.  Let $G$ be a monoid, written multiplicatively.  Define the additive group
\[ R[G]\colonequals \bigoplus_{g \in G} R = \left\{ \alpha = \sum_{g \in G} a_g[g] : a_g=0 \text{ for all but finitely many $g$}\right\}. \]
Define a product on $R[G]$ by $[g][h]=[gh]$, extending by distributivity.  Then $R[G]$ is an $R$-algebra, with multiplicative identity $[1]$ (check this if you need to!)\ called the \emph{monoid algebra} of $G$ over $R$.  
\begin{enumalph}
\item The polynomial ring $R[x_1,\dots,x_n]$ is a monoid ring: for what monoid?
\item Let $G=S_3$ and $R=\Z$.  Let
\[ \alpha=3(1\ 2)-5(2\ 3)+14(1\ 2\ 3), \quad \beta = 6(1) + 2(2\ 3)-7(1\ 3\ 2). \]
Compute $\alpha\beta$.
\item Let $f:G \to G'$ be a homomorphism of monoids.  Show there exists a unique $R$-algebra homomorphism $\phi:R[G] \to R[G']$ such that $\phi(g)=f(g)$ for all $g \in G$.  (Recall an $R$-algebra is a ring $A$ with an injective ring-homomorphism $\iota:R \hookrightarrow A$ such that $\iota(R) \subseteq Z(A)$; we usually drop $\iota$ and consider $R \subseteq A$.  An $R$-algebra homomorphism $\phi:A \to A'$ is a ring homomorphism such that $\phi|_R=\mathrm{id}_R$.)
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