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\begin{document}

\title[101 Worksheet \#6]{Math 101: Algebra I \\ Worksheet, Day \#10 \\ ``Rings extravanganza''}
\date{Tuesday, 27 September 2016}

\maketitle

\begin{problab}{1}
What is the noncommutative ring with the smallest cardinality?
\end{problab}

% upper triangular matrices in M_2(F_2) has size 8; all previous rings are necessarily commutative.

\begin{problab}{2}
Show that no commutative ring has additive group isomorphic to $\Q/\Z$.
\end{problab}

% the characteristic this ring is not defined.

\begin{problab}{3}
Let $R$ be the ring of all continuous functions on $[0,1]$.
\begin{enumalph}
\item Observe that the collection $I$ of functions $f \in R$ such that $f(1/3)=f(1/2)=0$ is an ideal.  Show this ideal is not prime.  
\item What are the maximal ideals of $R$?
\end{enumalph}
\end{problab}

% Easy verification for (a); for (b), the answer is that they are identified with the points $c \in [0,1]$ by $M_c=\{f \in R : f(c)=0\}$.  

\begin{problab}{4}
Let $k$ be a field and let $R$ be a (commutative integral) domain that is a finite-dimensional $k$-algebra.  Show that $R$ is a field.  Conclude that a finite domain is a field.
\end{problab}

% Let x <> 0.  Then the left multiplication map by x is k-linear, and injective because $R$ is a domain; therefore it is surjective, so there exists y in R such that xy = 1, and x is a unit.

\begin{problab}{5}
Let $F$ be a field, let $S$ be a ring, and let $n \geq 1$.  Show that any ring homomorphism $\M_n(F) \to S$ is either zero or injective.
\end{problab}

% It is enough to show that $\M_n(F)$ has no nontrivial two-sided ideals; this follows from a standard argument with matrix units.

\end{document}