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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 4.1  \\ Fall 2019 \hfill Due October 2}}}

\large

\vspace{1cm}

\begin{exercise}
Let $\varphi : R \to R_1$ and $\psi : R \to R_2$ be homomorphisms of rings. 

\begin{itemize}

\item[\bf a.]  Check that the product map $\varphi \times \psi : R \to R_1 \times R_2$ is a ring homomorphism.

\item[\bf b.]  Prove that $\varphi \times \psi : R \to R_1 \times R_2$ is a surjection if and only if (i) $\varphi$ and $\psi$ are surjections and (ii) $(1_{R_1}, 0), (0,1_{R_2}) \in \operatorname{im} \varphi \times \psi$.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $R$ be a ring and $\mathcal{I}$ a collection of ideals in $R$.  We say that the ideals in $\mathcal{I}$ are {\em pairwise coprime} if $A + B  = R$ for all distinct  $A, B \in \mathcal{I}$.

\begin{itemize}

\item[\bf a.]  Prove that if $A_1, A_2, \ldots, A_m$ are pairwise coprime ideals in a commutative ring $R$, then $A_1 A_2 \cdots A_{m-1} + A_m = R$.

\item[\bf b.]  Prove that if $A_1, A_2, \ldots, A_n$ are pairwise coprime ideals in a commutative ring $R$, then
\[
A_1 \cap A_2 \cap \cdots \cap A_n = A_1 A_2 \cdots A_n.
\]

\end{itemize}

\end{exercise}

\begin{exercise}
Let $A, B$ be a ideals in a ring $R$.  Suppose that $A \subset B$ and let
\[
C = \{ r \in R \, | \, br \in A \text{ for all } b \in B\}.
\]
Prove that $C$ is an ideal in $R$ containing $A$, and that
\[
BC \subset A \subset B \cap C.
\]
\end{exercise}

\begin{exercise}
Let $R_1$ and $R_2$ be rings.  For $k = 1, 2$, let $\pi_k : R_1 \times R_2 \to R_k$ denote projection onto the $k$th coordinate.

\begin{itemize}

\item[\bf a.]  Prove that $\pi_k$ is a ring homomorphism for $k=1,2$.

\item[\bf b.]  Let $I \subset R_1 \times R_2$.  Prove that $I$ is an ideal in $R_1 \times R_2$ if and only if $I = A_1 \times A_2$ for some ideals $A_k \subset R_k$, $k=1,2$.  [{\em Suggestion:  }Apply exercise 3.2.1 to the projection maps.]

\item[\bf c.]  Prove that $(R_1 \times R_2)^{\times} = R_1^{\times} \times R_2^{\times}$.

\end{itemize}
  
\end{exercise}

\end{document}
