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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 1.3  \\ Fall 2019 \hfill Due September 4}}}

\large

\vspace{1cm}

\begin{exercise}
Prove that a ring with a left cancellation law has no zero divisors.  In particular, a commutative, left cancellative ring is a domain.
\end{exercise}

\begin{exercise}
Let $R$ be a ring with no zero divisors.  Prove that for all $n \ge 2$ and all $a_1, a_2, \ldots , a_n \in R$,
\[
a_1 a_2 \cdots a_n = 0 \,\,\, \Rightarrow \,\,\, a_i = 0 \text{ for some } i.
\]
\end{exercise}

\begin{exercise}
Let $S$ be a set, $R$ be a ring and $a \in S$.  Prove that the {\em evaluation map} $E_a : R^S \to R$, given by
\[
f \mapsto f(a),
\]
is a homomorphism of rings.  What is $\ker E_a$?
\end{exercise}

\begin{exercise}
Let $R$ and $R'$ be rings and $f:R \to R'$ a function satisfying $f(a+b) = f(a) + f(b)$ and $f(ab) = f(a) f(b)$ for all $a, b \in R$.  In class I claimed that these conditions do {\em not} imply that $f(1_R) = 1_{R'}$ (which is why we included this additional equality in our definition of ring homomorphism).  Verify my claim.  That is, give an example of $R$, $R'$ and a nonzero $f$ for which $f(1_R) \ne 1_{R'}$. 
\end{exercise}

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